ASCAP and BMI for the US, and SOCAN for Canada.
I deal with this on a daily basis. I work for a company that provides our customers with a license ASCAP, BMI, and SOCAN so they don't have to worry about obtaining and paying for one. I used to work for Universal Records, now I work as a recording engineer/music programmer.
If you plan on performing using music from popular artists, they are most likely not royalty free music, and therefore you should technically have to pay ASCAP or BMI for the right to use it for commercial use (use while performing in public, or private).
Even if you have a license from ASCAP or BMI, it still doesn't give you the right to alter the song in anyway (editing). I know some performers out there edit they're music to fit their act, which is technically they need to get other permissions and rights, which can get pretty complicated as the Record label owns the rights to the recording, while the artist owns the rights to the composition.
Most venues SHOULD have a license to these organizations, such as bars or coffee shops if they have live music. I work in the music industry, and have heard of ASCAP and BMI sending letters to bars that advertise live music saying they need to pay for a license. They started cracking down on this a few years ago. If they do, and you perform there, you should be covered, but I wouldn't bet on it, since a lot of places still get away with not having one. Technically the live band or DJ should have a license, however, it's easier for them to go after the venue since they are the ones hiring such performers, and easier to track. A lot of people think, what's the big deal, I can play the radio in my store, after all, anyone can turn on the radio and hear the music for free. But all that is wrong, and they do need to pay if they want to even play the radio.
Basically the rule is, if you're playing music that is not royalty free, such as the radio, ipod, cd, etc, for or in your business, you have to pay ASCAP or BMI.
Now aside from all the technical talk, I think you'll be fine getting away with it. ASCAP and BMI will most likely not go after you if you plan on using it in your performances because they're not going to know (if you're doing private or even public shows). Just don't do it on TV and they won't know.