Obviously a complete record would be longer than one post, so consider this a series, assuming I find it valuable enough to continue, or if I get comments expressing interest.
What I had in mind to start is the King's Knight's Gambit, Fischer Defense, in a line that Fischer himself considered favorable (for Black) but uncommon in practice in his time: the 7...Ne7 line here (see commentary after 7...Nc6 in his main line) (pdf). Modern computer analysis shows that Fischer was absolutely right! So let's try to make this common practice and play it (1.e4 e5 2.f4 exf4 3.Nf3 d6 4.Bc4 h6 5.d4 g5 6.O-O Bg7 7.c3 Ne7):
The main idea, besides expediting the favorable O-O that Fischer mentions, is adding control to d5 while developing a piece, without putting the knight on f6 where it could obviously lose a tempo to e5.
The other easy question about Nec6 is "what about d5?", where the answer is pretty easy -- the knight finds a favorable exchanging square with Ne5. And if White tries b4 and b5, the knight is good with Na5.
The move suggested by Fischer at this tabiya is instead 8.g3. Here Fischer's analysis for Black seems to falter a bit. He's right that 8...d5 9.exd5 fxg3 10.Ne5 is bad for White due to 10...gxh2+. But he seems to miss that Ne5 a move later is still good for White: 10.hxg3 O-O 11.Ne5 (Fischer only gives 11.Qb3) Qd6 12.b3! and White beats Black to the punch with the Ba3 threat before Black can dislodge the knight.
The correct response to 8.g3 looks to be to hold onto e5 control, playing 8...Bh3 followed by 9...fxg3 instead:
So that's the end of my look "forward" from the tabiya. Let's step backward a bit. The main alternative I've encountered in games is 7.Nc3 instead of 7.c3. As Fischer also correctly says, that leaves d4 too vulnerable. Black plays 7...Nc6 immediately attacking it. Black's threat is to kick the f3 knight with g4 and take d4. If 8.d5, the same Ne5 idea above for Black is even stronger.