Just now I wanted to start a new game on an empty slot. Things went fine for a few minutes, then there was a glitchy moment and later still it stalled when I had reached a chest underneath the hut without coming back on. After I had restarted my computer all the slots with games that I had played before were gone.
That's strange - the saves should be in %APPDATA%\Fancy Fish Games\Aground\aground (assuming you're playing on windows), let me know the filesize of saves.sol (it should be many KB to 1MB depending how many saves you have and how far along they are). Also, keep in mind this is user-specific, so if you log in to another user account, that will have it's own set of saves (the first user's saves will be intact under their account).
Also, if you were using Aground Cloud, try logging in and it should re-download the saves.