In practice you will usually find that the player delivering stalemate has the most material and had the advantage a move or two before stalemate, depending on if there was a forcing way to cause it.
The way advantage is normally talked about, I would say neither player has it but it could be argued that the stalemated player has the advantage of not losing 
Well lets see if you have the heart for a stalemate or a heart of a winner, cause way i see it a looser run for a stalemate in advantage of not losing.
Nobody has an advantage. It's a draw.
Stalemate is a type of draw in chess that happens when the player who has the turn to move, doesn’t have any legal moves to make and there is no check on his king.
So, no further moves can be played in the game, hence a draw.
The position reaches a stalemate where it is one player’s turn to move, but his king is NOT in check and yet he does not have another legal move, Who has the advantage?
It's just a draw. No one has advantages.
The position reaches a stalemate where it is one player’s turn to move, but his king is NOT in check and yet he does not have another legal move, Who has the advantage?