I at least know that Wizards get to use level-0 spells an unlimited number of times a day, but is this a class feature, or are all cantrips unlimited use? I ask this because some spontaneous spellcasting classes still have a limited number of slots for level 0 spells, even if they aren't capable of learning that many spells of that level, and was wondering if they are limited use for those classes.
No, and wizards don't either.
Wizards learn all cantrips available to them automatically, but they are restricted by their prepared spell slots per day in cantrips as with every other level of spell. This slots-per-day limit is true of all other casters (spontaneous or prepared-type) with access to 0-level spells of any sort; the advantage wizards get is only in knowing all of the cantrips, not in being able to use them as often as they like.
[NB: Pathfinder did implement unlimited cantrips, which may be what you're thinking of.]