Many recreational players will show their hand even if they dont have to (showing what they folded/shoved/betted with, often on turn or river). Some will show only one card, often to mislead your analysis of the hand) I would never avocate for giving away informations. They're suppose to pay to get it. But in your opinion, would there be any situation where it is profitable ?
If so, wich ones and why ?

Classic situation is if you are a tight player, showing a three street bluff early in a session pays off dividends for hours. I also look early (yes, I'm tight as a tick) for medium suited connectors to PFR and then hit the flop hard. Actually verbalize after winning the pot

"Cmon guys, don't give me grief here. I raised the hand preflop with the virtual nuts....just like you guys."

If you're a tight player you're actually better off not showing your bluffs because you can better take advantage of your tight image. Showing value hands will confirm your tight image and will allow you to get away with well-timed bluffs in key spots. I would much rather have my opponents think I'm a nit who never bluffs than think I'm a tight player who has the ability to bluff.

I hardly ever show my cards but one situation where I will is if I can get info in return that is more meaningful than the info I give away by showing. For example, if I'm in a hand with a weaker player and he tank-folds to my river bet, I might show him my monster (which people will assume I have because of my image) if he is willing to show me his hand. The info I get from him is more valuable to me than the info I give away by showing my hand.

My thoughts exactly. One other time is when I'm playing with weak players who are frustrated at my game with comments like "Oh here he goes, raising again" etc. I'd rather they are happy than frustrated because I'd rather they are folding and respecting my raises and bets. So I'll show strong hands every now and then in the spirit of "But I'm not trying to rob you, I have the nuts" to keep them happy that their folds are well justified. Especially in home games and local tournaments where it's the same player pool each week.

The whole notion of 'not giving away information' only applies if you are so static that giving away information allows others to predict your future actions. Depending on the situation, I will show my hand

1. When I am playing counter to my normal style-If I am a tight player, I will show some bluffs or weak hands to try and widen the perception of my range
2. To improve my fold equity-If we are getting deep in a tournament, and I am having to shove often, I will show monsters to try and create the association that I bet only when I have it
3. As part of a metagame (I do this rarely)-If I am in a blood fued with a player, and I want to tilt him, I might show a bluff.

I am always aware of what i show, and I use that to gauge how it has affected my image.

In more challenging games where I am likely average or below average for the table, I don't show ever (except the harmless cases like when everyone folds to my BB, and I have AA)

It really depends on the players and the stakes.