2 c3 [B22]
The variation 1.e4 c5 2.c3 d5 3.exd5 Qxd5 4.d4 Nf6 5.Nf3 Bg4 was very topical some 10 years ago. The position arising after 6.Nbd2 Nc6 7.Bc4 Bxf3 8.Qb3! is rather amusing:
8...Na5! is the best response. In Howell - Chandler Black equalizes without difficulty but gradually loses the thread, and the game, against his young opponent.
In the old variation 1.e4 c5 2.Nf3 Nc6 3.c3 Nf6 4.e5 Nd5 5.d4 cxd4 6.cxd4 7.Bc4 Nb6 8.Bb5 dxe5 9.Nxe5 Bd7 White usually plays 10.Nxd7 if he intends to fight for a small advantage. In Acs - Kalinitschew Black introduces a new idea on move 12 in a well known position. Although White emerges from the opening with a nice advantage, I think Black missed a couple of chances to keep a level game.
3.c3 Sicilian [B50]
I am not sure what to call the 1.e4 c5 2.Nf3 d6 3.c3 variation, but it shows itself once in a while. We have not covered the common 4...g6 before. White's best try is Torre's 5.0-0 Bg7 6.Bb5+:
Black has several responses here, and while it is not clear what Black's best line is, they actually all look rather satisfactory! Kovacevic - Palac sees Black fighting for the initiative at a very early stage.
Another answer to the nameless system is the very solid 4...Bg4 which can lead to play similar to some lines of the Hybrid system of the Moscow variation. Kasimdzhanov - Anand is a good illustration of Black's possibilities and the game prompted Kasparov to say that Anand played on a "different planet" than Kasimdzhanov.
Closed Sicilian [B50]
Zvjaginsev uses an Anti-Najdorf move order to head into a kind of Closed Sicilian with 1.e4 c5 2.Nf3 d6 3.Nc3 a6 4.g3. The young Nepomniachtchi answers with 4...Nc6 5.Bg2 Bg4!? Which was an idea favoured by Bobby Fischer. Black gets a good game without any difficulties in Zvjaginsev - Nepomniachtchi.
Moscow 3.Bb5+ [B51-2]
The Hybrid variation has been popular lately, and there has been increased interest in 1.e4 c5 2.Nf3 Nc6 3.Bb5 d6 4.0-0 Bd7 5.c3 Nf6 6.Re1 a6 7.Ba4, which is generally less popular than 7.Bf1 or even the gambit line 7.Bxc6. In Smirin - Greenfeld Black plays the line 7...b5 8.Bc2 Bg4 which is thematic, but has always seemed a bit risky to me.
Ivanchuk takes up a ChessPublishing suggestion in the Hybrid variation 1.e4 c5 2.Nf3 Nc6 3.Bb5 d6 4.0-0 Bd7 5.c3 Nf6 6.Re1 a6 7.Ba4 c4 8.Bc2 with 8...Ng4!?:
However, he errs quickly in Svidler - Ivanchuk.
Three rounds later in the same event, Ivanchuk shows his great flexibility and tries the ambitious 1.e4 c5 2.Nf3 d6 3.Bb5+ Nd7. After 4.d4 Ngf6 5.Nc3 cxd4 6.Qxd4 e5 7.Qd3 h6 8.Be3 Be7 White usually plays positionally, trying for control the d5-square, but in Movsesian - Ivanchuk we look at the unusual 9.0-0-0!? Ivanchuk quickly gains the advantage after Movsesian's overly aggressive play.
Best wishes, David