I've got a lot of range, but I'm pretty picky. For a lot of things, I'll like a song or two by an artist and on occassion I'll like most of an album and a few songs from the artist's other albums 
I like Pop, Rock, some limited electronic and trance, some select Country, limited rap, just some dubstep so far (wub wub wub wub), and a lot of things that are lumped under Alternative. I like the more active pieces of classical and soundtrack music. Japanese pop and Anime music? Fuck yes.  
Oldies. Oh yes, I like oldies even though they're before even my time. 
Anyways, some of the bands and artists that have produced a lot of songs I've liked: 
Avenged Sevenfold 
Modest Mouse (Good News for People Who Like Bad News) 
Pink 
No Doubt 
Eric Clapton (Unplugged) 
Ayumi Hamasaki 
My Chemical Romance 
Elton John 
Nobuo Uematsu 
Paramore 
Rob Thomas 
Ronald Jenkies 
Santana 
Skillet 
Akira Ifukube 
The Wallflowers 
One Republic 
Paramore 
Wierd Al