French Advanced Variation:
Lately the Advanced line has not been very popular against the French defense, since it leads to close positions where white has a weak pawn on d4 and usually manages to exchange his bad light-square bishop. After the most principled move 6. a3 c4 black has a space on the queen's side and a very easy plan. Also 6.Bd3 and 6.Be2 lines don't offer white any advantage.
French Classical Variation:
The Classical line is the most principled and I believe the strongest response to the French defense. Currently many elite player opt for the classical line, when encountered with the French defense in the opening. White usually has more space advantage but black tries to put as much pressure as possible on the d4 pawn and try to exchange the bad light-squared bishop when there is an opportunity. If he manages to do that black is ok; if not usually black has to defend passively in a slightly cramped position.
French Exchange Variation:
Exchange variation of the French defense usually doesn't pose many problems for black , however black needs to play accurately to equalize in the opening. In symmetrical lines black is absolutely safe. The only line that offers white some active play is the one with 4.c4, with chances to get an opening advantage. Overall I would say that the Exchange line is relatively safe for Black.
French Tarrasch Variation:
Tarrasch line against the French is the most solid one and is a quiet position. Former world champion Anatoly Karpov used this line to score some fine victories in his career. Black needs to immediately put pressure on the d4 pawn and try to break thru with the f6 pawn move in order to be able to activate the light square bishop. In my opinion, one of the most memorable games in the Tarrasch is Reshevsky -Vaganian from 1976! I highly recommend all the French players to take a look at this fantastic game.
French Unusual Lines:
Unusual lines against the French defense are rare, yet once in a while you can find a top grandmaster who would like to avoid the main theory of French and try 2. Qe2, which was played against me by GM Vitor Laznicka - 2684 FIDE rating - at the recent World Open Tournament 2013! In 2012 the move 2.d3 was uses by none other than number one in the world GM Magnus Carlsen, to win a nice positional game against another top 5 player in the world: GM Fabiano Caruana.