Store brand karaoke mics which cost $15 and have the 1/8" plug on the end to plug into a karaoke machine are likely not going to sound very good, but this has more to do with build quality and parts than anything. They build them cheap to sell them cheap and the audience they are trying to reach isn't really concerned with sound quality. However, if this is what you have to record with then by all means go for it. This is exactly what I started out recording with and a radio shack condenser mic for drums. We all have to start somewhere. Working with "bad" equipment and squeezing every ounce of life you can out of it will help you hone your skills and you will be more able to appreciate the "better" equipment when you get it.
When it comes to live and recording, there is actually a lot of crossover between mics that are used live on stage and in the recording studio. For example, the Shure SM57 is a standard for guitar amps and drums both live and in the studio. Mics used for kick drum/bass cab will work the same both live and in the studio. Drum overheads will also work the same.
A main difference between recording/studio and live/PA is that condenser microphones are very common for vocals in the studio, but for live they are less common because condenser mics in general are more sensitive and thus are more prone to feedback. Microphones that work well live will also work well in the studio. Todd Wilcox mentioned Michael Jackson's Thriller being recorded on an affordable mic...this mic was the Shure SM7 / SM7b which brand new is $350 (extremely inexpensive compared to many studio vocal mics). This mic was actually created as a broadcast mic, but people loved it for studio and live so it kept getting used. The list of pro recordings that have used this mic is off the charts from Metallica, Incubus, Red Hot Chili Peppers, Smashing Pumpkins, John Mayer, etc., it's also used frequently live, and in broadcast the SM7b along with the EV RE20 completely dominate the field.
So really the main issue is whether the mic itself sounds good, not its price or what it's intended purpose was. The SM7b is not an expensive studio mic and it was not created for recording or live use, but it sounded amazing in these applications so now it's used all the time even by studios with essentially an unlimited budget...when a mic sounds good then it IS good! :)