I was at a woodworking show a few years back and came across a rep from Franklin Adhesives (Elmer's) who was selling melamine glue among others. I had a good talk about it and bought some. One of the most versatile glues I have ever used. It will glue melamine to just about anything else. The kicker is that one side must be porous so that the water base can "flash off" although it will glue small pieces of melamine to melamine because the water can eventually escape sideways. It is odd from other glues in that it is recommended that you do NOT clamp joints together. Brad nails are the best recommended practice, loose clamping next. I put some 1/2" birch sides on 5/8" melamine bottoms for sliding shelves and realized that I had made a measuring error and needed to take them apart. I split some of the birch doing it. It is a very rubbery bond. The biggest downside of using this glue is that NOTHING will take it off your hands. I've tried everything but toluene which I draw the line at.
If you are glueing the top sheet to plywood, etc., I recommend the melamine glue. If Melamine to melamine the contact cement might be better but you still have the same problem, the remaining solvent has to be able to escape through the gap between the sheets.
Someone I consider a master woodworker once told me that a master woodworker is not someone who never makes mistakes. He is someone who is able to cover them up so that no one can tell.