Wolodymyr Darmochwal, an active Plast member, gave this venture the moniker "Veselka", from веселка, the Ukrainian word for "rainbow."

It was able to expand during the economic recovery of the 1980s, at which time the row of phone booths at the rear of the restaurant came to be used as informal office space for East Village performance artists.

[10] The restaurant has attracted notable patrons including musician Ryan Adams,[11] artist Sally Davies, director Bart Freundlich,[12] performance artist Penny Arcade,[13] comedian Jon Stewart and actors Julianne Moore, Chris Noth, Parker Posey, Justin Long and Debra Messing (who considers Veselka her "late-night mainstay" and her "absolute favorite place").

Veselka continues to support the needs of neighborhood residents and Eastern European immigrants: in 1994, its kitchen staff included four doctors, three from Ukraine and one from Poland, who had recently arrived in the United States.

[18] After the 2022 Russian invasion of Ukraine, Veselka's daily patronage more than doubled, from 600–700 to 1,500, as many visitors wanted to express support for Ukrainians.

[31] Veselka is also featured in City of Fallen Angels, the fourth book in Cassandra Clare's The Mortal Instruments series.

Anthony Bourdain filmed an interview with publicist Danny Fields at Veselka that appeared in the final episode of Parts Unknown on CNN in 2018.

[38][39] Reviews of Veselka in traditional press highlight its comfort food menu and describe the restaurant as a destination for late-night diners.