Some friends got me into Andrew Jackson Jihad last year. While I wouldn't come right out and call them a favorite band, they scratch a very specific itch for me the way The Moldy Peaches and many Elephant 6 bands did back in the day. They're often political, occasionally profane, and they strike a great balance between funny and musically fascinating.
To narrow down to 20 songs can be a little problematic, but they're also a band that requires some curation (at least to start). This is probably a good starting point, but just know that they enjoy the naughty language.