Quote Originally Posted by Rondavu
-Raised pot flop brings your set and possible monster set.....

Check raise for info. You might save some money. You don't want your set of 3's being a one outer to a set of aces with you betting like you're god through all streets. Find out early whether they also flopped a set.

-Raised pot flop brings your set on a rag board....

Bet 1/2 pot to keep them in while looking like you have a cautious top pair. If they reraise then punch it against the high pocket. If they call then bet the same amount on the turn no matter what hits. Now it will be like 1/4 pot. The reason you don't raise your bet on the turn is because if a broadway hits the turn, nothing says made set like disregarding an over against a preflop raiser and betting hard. If another rag hits you still want him hanging around to pay you. Multihanded just take it down quick on coordinated boards.

-Unraised pot with at least one broadway and a draw possibility...

Min bet as if you're drawing, person holding broadway comes over the top. Smooth call and check to them. Let them try to chase you out with another big bet on 4th street. Check raise them. If the board coordinates take it down with a big bet.
For the first situation, a lot of players will check top set, so checking for the purpose of check-raising isn't likely going to be a very effective means of gaining information. Plus, many players will come over the top of a check-raise with a range of hands which are beaten by a set.

That aside, by check-raising with the intent of folding if your opponent shows strength, you're going to be limiting your profit in the likely situation where your opponent has top pair or an overpair since you're going to be slowing this player down, you'll save a decent amount of money in the hugely unlikely situation where you do have a lesser set, and you'll be losing money when you make a few erroneous folds due to your opponent overplaying a pair. So, overall, this doesn't seem like an optimal play since all it does is saves you a little money in a situation which is incredibly unlikely, while costing you money in almost every other situation. It seems sensible to instead concern yourself with maximizing profit in the likeliest situation where your opponent has top pair or an overpair rather than risk scaring these hands off because you're afraid of a higher set.