First off: the French Defense is a "correct" opening and you cannot expect to get some "automatic" advantage playing against it - the same way you can't expect to get an advantage in the Ruy Lopez, the Italian, or any other correct opening. That doesn't mean there aren't some lines which lead to either white or black advantage. But given best play from both sides you will end up in some balanced middlegame position.
Second: the main theme of the French is "counterattack". White is allowed to expand his center - only to have it attacked and eventually blown apart later. On the other hand black always has the light-squared bishop as a liability, because the pawn structure remains fixed for a long time and most black pawns are on the white squares.
There are some offbeat lines for white which are still "correct" in the sense that black can play correctly and still have no advantage automatically. The "offbeat" merely means that the resulting structures are unusual for the French Defense:
a) Chigorin Variation
After 1. e4 e6 2. Qe2!? black cannot really play d5 because the e-pawn, which would naturally take back in case of exd, is pinned and Qxd5 is uncomfortable because Nc3 would win a tempo. Black usually responds with 2. ... c5 and the game will develop very similar to the closed Sicilian, with the white light-squared bishop usually ending on g2 (otherwise white would have to lose an additional tempo to bring the queen out of the way). The resulting positions are about level.
In the main variation (1. e4 e6 2. d4 d5 3. Nc3 Nf6 4. Bg5 Be7 5. e5 Nfd7) The classic continuation is to exchange the Bishop on e7, but 6.h4!? is also possible. This is a gambit, but black's position becomes dangerous after 6. ... Bxg5 7. hxg5 Qxg5 8. Nh3! (hitting the queen) and a following Qh5. A typical motive is the sacrifice of a knight on e6, tearing open the black position and breaking through in the center. In the end the gambit leads to equality, not more (once again, the French is still "correct"), although the tendency of the game ending in anything else than a draw is high.
c) Tarrasch Variation
After 1. e4 e6 2. d4 d5 instead of the "normal" move 3. Nc3 (the "classical variation") white can also play 3. Nd2!?. on the upside d4 can be reinforced by c2-c3, the downside is that the bishop c1 is temporarily shut in. Black can steer the game into rather quiet waters by 3. ... dxe (the "Rubinstein variation") and it makes no difference if the white knight takes back coming from c3 or d2. Black can also play 3. ... c5 (the "open variation"), resulting in a lively game. Black can also ignore white's setup completely and play 3. ... Nf6.