In my understanding, S1 is the first Roadster, fitted with the 3.0 engine, 223 HP borrowed from the S type/Mondeo ST220, matched with the smooth Getrag 5 speed box. This is the the most rev willing engine in comparison with S2 and S3. S1 Engine has barrel tappets to get higher revs and the adjustment is a manual job. Built until 2006 or so.
S2 is from the US, matched with the Ford Transit gear box, 204HP and hydraulic tappets with automatic adjustment. Very relaxed drive, more cruiser than S1. S1 is a bit more lively but S2 also opens up nicely after long running in period. Built until early 2010.
S3 started with the Roadster Sport mid of 2010. Wider front axle than before. Some months later bespoke models became available with same wider front. Same engine as S2 but somehow tuned to 230 HP. Exhaust manifold got a slightly bigger diameter. Same Ford Transit gearbox. After more than 20K km the engine opens up and drives nicely. No rev hanging issues at all with S3 series which is a joy.
I must say, you can get used to the Ford Transit gearbox (S2/S3) easily, but in my view the ratios are not perfectly matching the power/weight ratio of the Roadster. First and Second gear are really too short.
So a 3.7 could be called an S4.
A big advantage of S1 to S3 is the comfortable legroom due to the narrower drive train tunnel in comparison with the Mustang 3,7 Roadster.
I'm wondering if some weight has been added with the appearance of the 3.7 Roadster? (more engine weight, servo steering etc?) I never drove a 3,7 but all 3.0 series.