Openings that Black has success with statistically? I love the B06 Modern Opening for Black:



[Title "White - Black"]
[FEN ""]

1. e4 d6 2. d4 g6 3. Nc3 Bg7 4. Bc4 e6 5. Nf3 Ne7 6. Bg5 h6 7. Bh4 a6 8.O-O Nd7



I find it leads to a strong position for White to attack and a secure one for Black to defend (especially if it leads you to the Hippo Defense because, as Black, I've had my greatest success with atypical openings:



[Title "Petrosian vs. Spassky, World Championship 1966 (game 16)"]
[FEN ""]

1.d4 g6 2.e4 Bg7 3.Nf3 d6 4.Be2 e6 5.c3 Nd7 6.0-0 Ne7 7.Nbd2 b6 8.a4 a6 9.Re1 Bb7 10.Bd3 0-0



Question: What are the major defences that statistically have seen the most success by Black?
It depends very much on what you intend by the word "advantage": if you are asking


  Are there any openings that give Black advantage at correct play?


the answer is obviously no, as well as there are no openings that give White advantage, at correct play. If otherwise there will just be no openings, rather only that one line that always gives advantage to White (or Black, respectively) without any need to play anything else. At correct play all standard opening variations are equal, if only there might be a tiny slight advantage for White in the sense that it is enforced to play the next move and as such the next-to-come threat.

If instead you are asking


  What are the major defenses that statistically have seen the most success by Black?


that is another matter completely. You might want to have a look at any of the large chess databases present on the internet where openings are sorted by winning percentages according to the variation (for the games contained in the database, obviously); an example is this page from ChessTempo showing winning percentages for the Sicilian Najdorf variation, for instance, that is well-known to be among the most played defenses versus 1.e4. Other examples for 1.d4 are the Slav defense or the Indian defenses, as standard sound counterplay for Black.