Here's my list:
---Super low action and light strings. I'd rather there be a slight fretbuzz than sacrifice higher action. So I get every bass I have as low as I possibly can, and never use anything heavier than 40-100 or 40-120 for 5 string, even when playing flats.
Yep. Same exact thing here, although I can go with heavier strings. I prefer 45-105 for rounds and TI jazz flats for... well, flats!
---Neck radius. Between 9.5" and 12" seems to be my comfort zone for a 4, though I just got a Yamaha 5'er with something like a 24" radius and it feels good. But any rounder than 9.5" and I can't bend strings without choking out or raising the action. And that ain't happenin'.
In the past I was very particular about that. Frets also. I wanted the flattest board and XJ frets. Nowadays, I'm fine on anything, although I do prefer 9.5" and shorter frets. Still, with my action preferences, 9.5" ALWAYS chokes if I bend. So I don't bend...!
---Light weight. I wear the term "weight weenie" like a badge of honor and don't care if you he-men "Just go hit the gym" types like it or not. Wasn't always a weight weenie but a chronic should problem, due to a heavy-as-heck Jazz I used to own on a 2" seatbelt strap, put a stop to that. I've got a limit and I do my best to stick to it, and since I've got light basses that I love already, I see no need to buy any that are going to make my shoulder worse.
I used to not care at all about weight. Then I got a jazz that was about 6kg. It got tiresome at gigs but still, it wasn't much of a dealbreaker, really. Then, I'd just -prefer- a lighter bass. A few months ago I discovered I've got multiple herniated and bulging discs in my mid and upper spine and last month I had ACDF surgery. After that, the only way I'm playing a heavier bass is sitting down. I still can't play standing up with anything heavier than an acoustic and even that's kinda stretching it...
---Black. If it's not black, it must go back. I don't mind other colors interspersed with the black base color, but a bass has to have black as the base color for me to buy it these days since I scientifically proved that I look the best with a black bass. Any bass you might see in my little collection that's not black was grandfathered in before the testing.
For years I've had a strong aversion to black instruments because they were so damn common, especially in metal that I'd gladly play death metal with an orange flake bass. Now I won't really reject a black bass and I'd love an all black one but still, I tend to prefer other colors.
One extra thing I have is neck width and girth. Once again, I'm not picky enough to treat this as a dealbreaker but I prefer a 40-41mm nut width instead of both the standard JB 38mm or the ultra wide P 43-44mm width. On four stringers.
As for the girth, I LOVE a round, fat neck. It's just so comfortable for me. I don't have big hands and I'm not really a big guy but those baseball bat necks are just so damn cozy! Think early German Warwick necks! I'm fine on thinner stuff too, even Ibanez toothpick ones but still...
String spacing is another thing. Only thing that's a dealbreaker, really. Can't stand more than 19mm on a four strings and I will absolutely reject any five string with 19mm at the bridge. I love tighter spacing both on four and five (or six) string basses. If I didn't think it'd look silly compared to the width and taper of the necks, I'd equip all (well, both) my four strings with Schaller 3D bridges and drop that spacing at least one millimeter.