Are you really getting better hands to fold?
4 things could happen imho.
You bet they all fold anyways
You bet somevillian calls and more often than not your beat.
You bet somevillian re-raises, you probabaly gotta fold.
You check, you still gotta shot at winning the hand.

So betting here acheives very little.
As above, agree with checking back being the best option.

Your hand has showdown value and your value is super thin.

Just check behind and rake in the moderate spot.  I can't really think of many hands that would hero you here.

I don't see any reason to bet.  I have played at the mirco level for way to long and know that people are going to call you way to often and possibly with better 10s.  There really just isn't a lot that you beat here.  Showdown value is really all you have going here.  If there was only one player maybe you could come with some reason for value betting the river but I'd have a hard time finding any. 

The deciding factor for me in these situations is the information I gain by checking and seeing their cards. I am going to get info on the ranges of the 2players to my immediate left and how they play from the blinds. I think you can argue those are the 2 most important players at your table. The information you gain on their blind defense will allow you to play optimally on future buttons.