Chocobo Races - On the more recent Windows operating systems, going through the Chocobo Races will crash the FFVII program. An unofficial patch was made that addresses this problem, which can be downloaded from this link. Simply unzip the patch and run it, and it should fix the problem.
Cosmo Canyon FMV - During the scene in Bugenhagen's observatory, the game seems to crash with extreme regularity during the FMV, right after the shooting star goes by. The cause and fix of this crash is, at the current time, unknown. However, the crash can be bypassed by sending your save file to someone whose game is unaffected by the glitch, and having them run through the scene and send the new save back. The rest of the game may then be played as normal. I'm able and willing to help people through this section, so contact me via PM if you experience the issue.