Barrows' hat, with a veil attached
A veil was a long, gauzy item of clothing usually worn on or over the head.
When reveling in the beauty of the Shore Leave Planet, Yeoman Tonia Barrows thought she ought to be wearing a tall hat with a veil. (TOS: "Shore Leave")
Veils could also be worn when in a state of mourning. (TNG: "The Nth Degree")
A veil could also be used as a metaphor for an obstruction to understanding. While praying to the Prophets, Kai Winn asked them to lift a veil of darkness from Benjamin Sisko. (DS9: "Rapture")
A Risian traditional wedding dress that Leeta was shown in 2373 also appeared to feature a veil. (DS9: "Call to Arms")
The exocomp Peanut Hamper wore a flowered veil and a false beak during the ceremony of her wing-joining to Rawda. (LD: "A Mathematically Perfect Redemption")