The basic Jerome Gambit position, after 1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+, appears in about 11,600 games. White scores 45%
The Italian Four Knights Jerome Gambit, 1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Nc3 Bc5 5.Bxf7+, appears 2,300 times. White scores 38%.
Interestingly, the Semi-Italian Four Knights Jerome Gambit, 1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Nc3 h6 5.0-0 Bc5 6.Bxf7+, appears only 120 times, but White scores 65%. Note the "power" (or lack of) ...h6.
This is reflected in the Semi-Italian Jerome Gambit, 1.e4 e5 2.Nf3 Nc6 3.Bc4 h6 4.0-0 Bc5 5.Bxf7+, appearing in about 160 games, where White scores 66%. Interestingly, with 4.Nc3, in about 130 games, White scores only 54%.
The Blackburne Shilling Jerome Gambit, 1.e4 e5 2.Nf3 Nc6 3.Bc4 Nd4 4.Bxf7+, appears in about 5,000 games, with White scoring 56%.
Recent examples of the Abrahams Jerome Gambit, 1.e4 e5 2.Bc4 Bc5 3.Bxf7+, have been added. There are a little over 2,500 examples, and White scores 50%.
Because the line continues to show up, and is worth knowing, the nameless 1.e4 e5 2.Nf3 Nc6 3.Bc4 Na5 4.Bxf7+ appears in The Database 2150 times, with White scoring 72%.
There is also a large selection of Jerome-ish lines, featuring, for example, an early Bxf7+ by White. Some Reversed Jerome Gambit games have been included, as well.