(written from a Production point of view)
Star Trek III: The Search for Spock, was one of the few Star Trek productions that introduced a multitude of new space faring designs at once, six in this case, including the Oberth-class. Though it was introduced as the fourth new Starfleet vessel design in Star Trek III: The Search for Spock as the USS Grissom, a far greater restraint has been observed, in comparison to the contemporary Excelsior-class studio model, when calling upon the Oberth-class design for repeat performances in later spin-off Star Trek productions. The design had only a limited presence in the Star Trek films, Star Trek: The Next Generation, and one appearance only in Star Trek: Deep Space Nine.
For The Search for Spock, a unique approach to designing was adopted, not seen before or after in the Star Trek franchise. Instead of the traditional way of thinking out a design, devising a design, coming up with detailed drawings to be approved of by visual effects supervisors and building models from blueprints, this time visual artists David Carson and Nilo Rodis-Jamero of Industrial Light & Magic (ILM) produced their pre-visualization artwork and handed it over to model makers Steve Gawley, Bill George, and their team to be translated into study models, in essence inviting them to use their own imagination to finish up on the design. Very much a collaborative effort, Carson later remembered, "We'd churn out quite a few sketches. Then the ones that were most promising we might polish up a little in color for presentation. It wasn't uncommon for me to do a drawing that would inspire Nilo, who would then turn it into his own drawing that would be much more impressive! He would often inspire me." (Star Trek: The Magazine Volume 3, Issue 8, p. 48)
Once turned over to the model makers the resulting study models were presented to producer Harve Bennett and/or director Leonard Nimoy for appraisal or as supervising model maker Gawley put it, "You had all these models sitting on a table so that the director could really get a feel for what we were talking about. It just made everything easier to understand, and insured (sic) that everybody was on the same page. It also made it easier to give cost estimates." (Star Trek: The Magazine Volume 3, Issue 8, p. 20) Director of Photography Kenneth Ralston elaborated further, "From the beginning, once Nilo Rodis, one of the visual effects art directors, had done a sketch and they got an idea of what direction to go, the model people all built prototypes. The space dock had four or five small prototypes. The Bird of Prey, I think, had only two because we all knew this one design would work and we were selling that one. The Merchant Man and the Grissom also had several designs. When Leonard and Harve and Ralph Winter came to meetings we presented them with three dimensional models. It really is a lot better doing it that way because they can physically see how different angles would work." (American Cinematographer, August/September 1984, p. 62)
The Valiant study model
|The Valiant study model|
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One of the study models Ralston referred to was labeled as the Valiant, and was featured in the 2002 special feature, "Space Docks and Birds of Prey", on disc 2 from the Star Trek III: The Search for Spock (Special Edition) DVD, already very close to the final design. USS Valiant was originally the name given to the vessel in the first draft script of The Search for Spock, before it was decided to rename the vessel after astronaut Virgil Grissom who had perished in the tragic Apollo 1 fire at Cape Kennedy in 1967. (Star Trek III: The Search for Spock (Special Edition) DVD–text commentary)
Photo documents reportedly taken around the time of the filming of Star Trek Generations of this Oberth-class model have given rise to some speculation that it was this model that was used in the evacuation scene of Veridian III.
However, screencap analysis showed that the layout of the impulse deflector crystals as well as the layout of the impulse engine exhausts/transporter emitters on the nacelle deck did not correspond with the layout as shown on the actual study model. Furthermore, study models were never outfitted with an internal lighting system as that constituted a flippant expenditure of budget funds. It can therefore be surmised that it was in fact the original physical studio model, still wearing the USS Pegasus decals.
Physical studio models
The original studio model for the Oberth-class, built at ILM by Gawley's team, measured an overall length of 28 inches. The model first appeared in The Search for Spock as the USS Grissom (NCC-638), and was subsequently relabeled to represent other vessels of the class. The first time the model was relabeled was for Star Trek IV: The Voyage Home as the USS Copernicus (NCC-640) as recalled by ILM's Model Shop Supervisor Jeff Mann, who has stated, "We had an incident in the beginning of the film, where we needed a Reliant-class [sic.], so we put a new paint job on the old Reliant model, changed a small shuttle [sic.] called the Grissom to the Copernicus and we added a back half to the shuttlecraft that Scotty flew around in Star Trek: The Motion Picture." (The Making of the Trek Films, 3rd ed., p. 68)
Neither name nor registry was ever discernible on screen, but were carried over to its next use as the SS Tsiolkovsky, when it was filmed at Image G for the The Next Generation's first season episode "The Naked Now". The fourth movie's registry still being there was later confirmed, when the remastered version of the episode was released in 2012.
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Michael Okuda revealed on that occasion, "I seem to recall that Grissom may have been relabeled to serve as another ship (the Copernicus?) in Star Trek III or IV. I didn't try to relabel the model for "The Naked Now", partly because we realized that the existing registry would not be legible in standard-def video, but also because we were all so insanely busy at the time that no one could take on an additional project that wasn't likely to be seen on the screen."  The number was digitally changed to its correct one, NCC-53911, in the first full side view establishing shot. Unfortunately, the digital artist overlooked the previous scenes and the later scene when the stellar core fragment smashes into the Tsiolkovsky, as it there still carries the original, now discernible, registry number.
Subsequently, for the television franchise, the model was relabeled trice, usually done at Okuda's art department where Doug Drexler was one of its staffers assigned the chore, first as the USS Cochrane (NCC-59318) for The Next Generation's fourth season episode "The Drumhead" (stock footage of which later on used as the USS Biko in the sixth season episode "A Fistful of Datas") and for its only Deep Space Nine appearance in the pilot episode "Emissary", albeit as two separate class incarnations, firstly as the ill-fated USS Bonestell (its name and registry – NCC-31600 – not discernible though) and secondly as the Cochrane again. Since the model made an encore as the latter, it is most likely that it had not been relabeled as the Bonestell due to its very fleeting appearance shortly before its on-screen destruction, very much akin to Okuda's earlier decision not to do so either with the Copernicus/Tsiolkovsky. It should be noted that all motion control photography for the model in its television appearances were done at Image G, whereas ILM was responsible for the model's photography for all its movie appearances.
The second time the model was relabeled as the USS Yosemite (NCC-19002) for The Next Generation's sixth season episode "Realm of Fear". At that occasion neither name or registry was ever discernible on screen, however they could be gleaned from the April 1993 cover of the Cinefantastique magazine, which was devoted to Deep Space Nine. The cover artist, David Voight, incorporating the registry number, had an actual behind-the-scenes photograph of the studio model, dressed as the Yosemite, at his disposal as reference.
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The model was relabeled for a last time for The Next Generation's seventh season episode "The Pegasus" as the USS Pegasus (NCC-53847),  with subsequent appearances shot in such a way that the decals could not be discerned. Very little further modification, save for some paint touch-ups (possibly to repair some minor damage after Jein took molds from it) was ever performed on the model besides relabeling. The last appearance of the model was in the Veridian III evacuation scene in Star Trek Generations.
The model, still labeled as the USS Pegasus, was auctioned off as Lot 706 on 7 October 2006 as part of the 40 Years of Star Trek: The Collection auction. The estimated price for the model was US$4,000 – $6,000; it sold for US$18,000 ($21,600 with premium). The model was added to the collection of ScienceFictionArchives.com, an European organization that is dedicated to preserve science fiction production assets for public display purposes, such as in museums. 
The SS Vico model
A second physical model was later created at Gregory Jein, Inc. for The Next Generation's fifth season episode "Hero Worship" to depict the severely damaged SS Vico. At the time considered too valuable to cut up the original ILM model for showcasing the damaged Vico. Gregory Jein took molds from the original ILM model and from them cast the parts to construct the new model. Co-worker Bruce MacRae helped out with detail construction and painting. 
Unlike its progenitor, the Vico model itself, having escaped the 40 Years of Star Trek: The Collection and It's A Wrap! sale and auctions, is still in the possession of CBS Consumer Products and has been on tour displays such as Star Trek World Tour, Star Trek: The Adventure, and Star Trek The Exhibition as late as 2012.
Jein later gave away the molds to Star Trek co-worker and personal friend John Eaves who actually used them to cast a solid resin model and had it gold-plated.  It, along with other starship models, was intended for use as a display piece in the glass cabinets in the observation lounge of the USS Enterprise-E in Star Trek Nemesis. However that fell through as Eaves delivered the models too late for use.  As Eaves himself recalled, "For the films, I was commissioned to build these models and have them gold plated at Artcraft Plating in Burbank, to be featured in the set case. Over the span of the three films, First Contact, Insurrection, and Nemesis I had to build a lot of models. For Nemesis the list grew with the request to build and plate some additional models. The Voyager, the Grissom, the Excelsior, and the Reliant. We had two cases to fill and these models were to accompany the Enterprise models. The set was constructed in an arch with a display case on either side of a large video playback display. The Enterprise models were to be on the right side and the new non Enterprise models were to be on the left. While the new ships were being plated the shooting schedule changed and the set had to be used sooner than later, because of this a double set of the Enterprise models were set in both cases. The new models barely made it back in time, but due to the amount of work involved to change them out it wasn't worth the effort, so in the end the new ships were ditched for the two sets of the Enterprises." 
Eaves retained the models, "I got the other ships done in about a week and took them to the stage and he [John Dwyer] had already filled the cases twice with doubles of the original 6. It was too much work to take out and replace the dups with the new so they didn't make it on the big screen, but they did wind up at my house again.".  Eaves, citing "tough times" as reasons for reluctantly doing so, eventually sold off his gold-plated Grissom model at eBay on 3 July 2013 for US$1,900. 
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The last appearance of the Oberth-class was in Star Trek: First Contact in the Battle of Sector 001 scene. With the exception of the Borg cube, Borg sphere, the unnamed Nebula-class starship, and the USS Enterprise-E, all other ships were executed as CGI models, including the at least three Oberth-class starships. Never meant to be seen up close but rather serving as deep background elements, the CGI model was built at ILM by modelers Larry Tan and Paul Theren, using Electric Image software for animation and Form-Z software for the model, at a fairly low resolution and at a low detail level. Noteworthy was that this was the first and only time that viewers could see an Oberth-class vessel discharge it weapons.
Established as an older design, superseded by the Nova-class as far as production staffers were concerned, neither the physical studio model was ever used again, nor was its CGI-counterpart ever upgraded for potentially later appearances, and the class has not been seen (though referenced to) in either Deep Space Nine, with the sole exception of the use of the physical model in the series' pilot episode "Emissary", or Star Trek: Voyager.
Still, in 1997 a more detailed CGI model of the USS Grissom was constructed in the LightWave 3D software at Foundation Imaging by Robert Bonchune, for representation in officially licensed print publications. This model was first featured as File 31, Card 7 in the 1997 issue 37 of the Star Trek Fact Files partworks publication, and reused in its 2001 US magazine derivative, Star Trek: The Magazine Volume 2, Issue 1, p. 34. In 2006, Bonchune's former Foundation colleague Adam "Mojo" Lebowitz again re-utilized the model as part for his contribution to the 2007 outing of the Star Trek: Ships of the Line calendar series, where it was featured on the center spread, depicted in the Battle of Wolf 359.
In 2014, Bonchune revisited his build when he was asked to re-render his model for a far more prominent and detailed representation in the Star Trek: The Official Starships Collection partwork where it was slated to receive its own dedicated issue.
- Star Trek: The Official Starships Collection, issue 36, December 2014