During USATT sanctioned tournaments you can only play with USATT approved rackets (paddles). USATT relies on ITTF (International Table Tennis Federation) rules for approved equipment. Here are simplified rules to check if your racket is approved (legal to use in tournament)
- The racket has to be red on one side and black on the other. Other colors are not permitted. Here is a complete list of these rubbers approved by ITTF.
- The blade (the wooden part) has to be at least 85% wood, the rest can be carbon fiber or paper.
- The colored rubber and a optional sponge underneath together can’t be thicker than 4 mm.
- The rubber can have some damage around the edges as long as it is not more than 2 mm from the edge of the blade.
To find out if your racket is legal just look for the ITTF logo on the bottom part of the rubber (close to the handle). 
Here are some pictures of this logo on the actual rubber.
fvttc.org 