While it may seem shallow, looks are important. Especially something a little out of the ordinary (I have allowed two FSOs into the house, one Variax and one Sei). Part of this is that it should be a natural finish, I dislike paint, especially as I'd probably chip it.
Number of strings - five (I have two wonderful 4-strings and more 6-strings than I'll ever use).
Headless preferred but not absolutely essential, though the threshold of acceptance for a headed bass is higher than that for a headless, obviously.
Once I've got that far, it's time to play it. Neck must be shallow. Not bothered about string spacing, anything from 16 to 19mm will do. Preferably two pickups and active, though I do have one passive and one single pickup bass. It should also make a reasonable sound.