Hey, I think I’m qualified to weigh in, given my user name, my music collection, and my experience DJing swing dances and playing sax in a ska band in the late '90s.
Mighty Mighty Bosstones: As said earlier, “The Impression That I Get,” from the album Let’s Face It, was their big hit. They followed it with the single “Royal Oil,” but it got a lot less radio airplay. Before that album, I guess around 1995, the band had a brief appearance in the movie Clueless, playing “Someday I Suppose” at a party scene.
The only other ska bands that attained national prominence during that time were Reel Big Fish, with their big hit “Sell Out,” and Save Ferris, the girl-fronted band who did “The World Is New.”
As for swing, Cherry Poppin’ Daddies’ only real big hit was the aforementioned “Zoot Suit Riot” (from the outstanding album of the same name), and Royal Crown Revue had a minor hit with “Hey Pachuco!”, which they performed in the Jim Carrey movie The Mask in 1994. (RCR is my favorite of all the late '90s swing bands, but they were never quite as commercial as Big Bad Voodoo Daddy.)
Big Bad Voodoo Daddy was a little bigger. Of course they came to prominence with their appearance in the 1996 movie Swingers that brought the neo-retro swing culture to the mainstream for a short time, and made stars out of Vince Vaughn and Jon Favreau. Their hits, both from the movie, were “Go Daddy-O” and “You and Me and the Bottle Make Three Tonight.” Those songs appeared on their self-titled album and also the Swingers soundtrack. They had a lesser hit with “Cruel Spell,” a slower song which appeared in a few TV shows and movies.
The Squirrel Nut Zippers combined swing with Dixieland, old-timey ragtime – they didn’t go for the whole zoot suit thing, and featured fiddle, banjo, lower brass, and a girl singer who sounded like a cross between Billie Holiday and Betty Boop. Their big hit, again, was “Hell” (with the guy singer), which they followed with “Put a Lid On It” (girl singer). A few years later, they tried again with “Suits Are Picking Up the Bill” from their third album, Perennial Favorites, but it didn’t get big.
Finally, Brian Setzer, former singer/guitarist of the '80s rockabilly revival trio known as the Stray Cats, fronted the Brian Setzer Orchestra in the late '90s and continues to lead them today. They had a big hit with a cover of Louis Prima’s classic “Jump Jive and Wail,” which was featured in a Gap commercial around that time, with some excellent swing dancers. (Or was the original in that commercial? I forget.)
Is this helpful at all? I can get waaaaay more obscure, but I named anything that would have had mainstream exposure (radio airplay, music videos, or soundtrack spots).