I am currently watching the UEFA EURO 2016 matches and I saw the referees checking the shoes of the players.
What are they looking for?
The check at the start of the game is a simple safety check.
There are no specific rules governing types of footwear except that they "should not be dangerous".
In practice this means that the ref is looking for:
Note: I'm aware of how ridiculous this all looks in the context of a major competition, it really is a matter of the ref just ticking the boxes. At an amateur level however, these problems occur regularly enough (especially the first point) that it's worth checking.