ARook is short stacked in this hand. I'm going to try to shrink this to something like hand converter for all the loyal people who have put up with my crap formatting.

***** Hand History for Game 13393617 *****
0.10/0.20 Texas Hold'em (Split Limit)
Table Seed (10 max) (Real Money) -- Seat 7 is the button
Antonella16 : Small Blind ($0.05)
Ray_Powers : Big Blind ($0.10)
macchine1 : Big Blind ($0.10)
** Dealing down cards **
Dealt to Ray_Powers [ As Ad ]
LUCASSY Call, Fold, ARook Call, Check, stevepar78 Call, Fold, Ray_Powers Raise, LUCASSY Call, ARook Call, 2 Folds
--- Nothing amazing here. I have pocket aces. I want the most money I can get for them.
** Dealing Flop ** : [ 3s Jd 9s ]
--- This looks fine to me. The flush draw is there which is slightly scary, but I don't see anyone betting this, so I bet for callers.
Ray_Powers Bet, Fold, ARook Call
** Dealing Turn ** : [ Jh ]
--- Well, that's kind of scary. If ARook was calling with aj or something, I am now in trouble. I think I still have to bet here. If he's on the flush draw, I need to make him pay to draw to it. If he raises me, he probably has a jack.
Ray_Powers Bet, ARook Call
** Dealing River ** : [ 5s ]
I hate you spades. I hate you to death. So, do I check/call here? I think I can't fold, but I'm not sure betting was right.
Ray_Powers Bet, ARook Raise DOH!, Ray_Powers Call
** Summary **
Main Pot: $2.15 | Rake: $0.10
Board: [ 3s Jd 9s Jh 5s ]

I check call here.

The flush draw beats you, the lone J beats you, a random middle/high pair will fold if you bet, but probably bluff the scary river if you check. A solid player could raise you with any 2 card, tho at those limits, they are not legion.

Yeah, check/call is probably best here.

I wasn't worried about the jack, as I figured the jack would have raised me on the turn, unless I actually ran into a tricky player at such a low limit game.

I think check call may have been right tho.

Ray

I think a check/call is terrible here. This is 10/20 cent, I think you're missing out on soooooooooooo much value not betting this river. Most of the time this is a 3, 9, some random PP, or A-high. And even if those hands aren't there >50% of the time (I think they are though), betting is still better than check/calling. He needs to have you beat here a lot for check/calling to be correct.

10/20 cents doesn't mean nobody ever hits anything because they're lesser players. Even considering the size of the pot, we're beat here too often not to check/call. Moreover, if villain does have a lesser hand, you can induce a bluff from him when he would have folded to a bet.

Yes, but the range of hands you figure him to call is huge since it's 10/20 cent. Further, since it's 10/20 cent, we can assume villian is LAP, which means you won't be inducing any bluffs.
When playing OOP on the river headsup vs a loose and passive player, check/calling is pretty much never the right thing to do. Their calling range is like 5 times as big as their betting range, so betting adds a lot more EV (although it still might end up being -EV itself).