if you had these positions what piece(s) would you prefer to have on the board to improve your chance of converting a pawn advantage (your opponent gets the same piece(s)).

It depends as much as on the location of the pieces, as which pieces they are. Am I allowed to choose the squares as well?

as long as a piece isn't being forced into passivity or there are direct tactics or the pieces would be forced to trade it's more of a general thing

diagram 2 with bishops on the board is likely winning for the side the pawn up, the other positions are harder to call. and of course having another piece on the board makes it murkier but your opponents aren't going to let you trade off pieces when they are down a pawn so i want to know what pieces i want to trade if i had the choice in an endgame

what the hell dude step off, i just wanted to ask a question not have someone nit pick what i say

maybe there was nothing nasty in your previous comments but you are starting to now but nit picking on the fact that i said "letting" is completely unnecessary and in a way derails the thread 

Probably knights in most without analysing the specific positions, they seem to be the winningest, it's hard to coordinate a defence.

I could win 1000 times in 1000 with any other piece (excluding a pawn). The only time I might lose in any is a mouse slip, where all pieces have the same problem.

Perhaps you didn't read that your opponent would have the same piece?

Heh, oops, nevermind. I mean, yeah, I knew that, I'm just that good. :)

This question is sorta vague unless you give specific squares for the pieces to be placed on...

btw, are we allowed guessing what opening these pawn structures arised from  I'm gonna go with Benko Gambit for #1 and Adv French for #4 

(Paraphrasing) "Knight endgames are pawn endgames" -Botvinnik

A pawn up, everything else being equal, knight endgames are the least forgiving, so in general I'd take knights.

Generally speaking opposite bishop endings and rook endings are the most forgiving, but of course a lot depends on where the pieces are placed.  I might venture a guess that (provided the kings have cover) queen endings are less forgiving than say bishops of the same color.

When placing and deciding what pieces would allow a draw, one feature in the position to look at is white's weaknesses i.e. how black can generate play i.e. how active the piece can be.  In all these positions (except #2 and #4) the base of white's pawns are still on the 2nd rank (harder to get to).  Also all the positions (except #1 and #2) are symmetrical except for the extra pawn so that the weaker side doesn't really have chances to create a passed pawn.

And on the subject of piece activity, the kings matter too.  In all the positions neither king has an advantage.

So in the positions where one side has an extra pawn, other than #2, #1 offers the best drawing chances because b3 may be a target and if lost black's c pawn would be a passer. 

#4 offers good chances to black because other than being a pawn up the d pawn may be targeted.

I hope that was general enough to answer what you were thinking of.  For more specific information I suggest taking the pawn structures from late mid game to endgame of real games and asking about those.  This would feature pawn weaknesses and a difference in king activity and would give more practical and specific answers.

what is the best piece to convert a win a pawn up

Oops, my answer turned out to be really long.

The simple answer to your title is simply (in the majority of practical cases) "the knight"

thanks waffllemaster for your insight

The knight, as noted above.