Because modern Rock, Pop, and Dance music is based on the Swing/Jazz/Blues lyric structure and rhythm, you can pretty much swing dance to any modern song that has an appropriate tempo.
When it comes to swing dancing, you really are talking about either West Coast Swing or Lindy Hop. Lindy Hop, as the original dance from the 1930's, is meant to be danced to faster music with more energy, while WCS is a smoother variation that changed as the music changed.
To compare, here are two videos.
West Coast Swing, as it's danced today, to the music that normally goes with it:
http://www.youtube.com/watch?v=V3ZxiPKmacg
Then, here is Lindy Hop, to music that Lindy Hoppers normally don't dance to:
http://www.youtube.com/watch?v=nYsW7vsXEX8
Obviously, this is fun, but normally Lindy Hopper would rather be dancing to something like this:
http://www.youtube.com/watch?v=pdgaVz56hlA
So I guess you have to ask yourself, which dance do you most identify with, and what kind of music are you going to like the most. If you don't like the music that's played most often, then you won't enjoy yourself, so keep that in mind.
So to answer you question....
If you are talking WCS, then some songs would be like:
"Bringin' Sexy Back" - Justin Timberlake
"Toxic" - Britney Spears
"Beggin'" - Madcon
"Idlewild Blues" - Outkast
"The Way You Move" - Outkast
"The Windmill" - The Gorillaz
Just to name a few.
If you are talking Lindy Hop, then really the heart of the music is with real swing music, but there are a lot of amazing modern swing bands recording songs today. Probably the best would have to be "Jonathan Stout and His Campus Five" who the home town favorites in the LA swing scene:
http://www.amazon.com/gp/product/B000QJQMPW/ref=sr_shvl_album_1_artist_rd?ie=UTF8&parent=B000QQWR68&qid=1261425925&sr=301-1
Check out their versions of "Diga Diga Doo", "Bugle Call Rag", "Dark Eyes", "Flying Home" and "Minor Swing" just to name a few.
Some other great bands would be:
Bill Elliott Swing Orchestra
George Gee Big Band
Casey MacGill and His Spirits of Rhythm (or his Blues 4 Trio)
Indigo Swing
Stompy Jones
Anyways, I hope that helps!