The answer is it all depends where you are in the world.
The OP lists singapore (I don't know there), Julian assumes UK, (where I have an idea there's no 5700kg limit on SEP land class, though I may be wrong)... and out here in skippyland, it's fixed pitch, fixed gear <5700kg, and you need a rating for CSU / Retract etc., not just differences training. However, you can apparently fly a centreline thrust twin on a SEP rating, whereas the UK requires differences training from a non-centreline thrust twin to a centreline thrust twin.
In short, it's all extremely confusing and not very well standardised, but you *should* be fine in pretty much any fixed gear, fixed prop single. To get a comprehensive answer you'll need to specify under what aviation authority you're flying, and probably dig out their set of rules.
Oh, and just an edit to add: As far as I'm aware most parts of the world give you a Single Engine PISTON rating - that precludes turbines. Even if it didn't, I'm pretty sure most people operating turbine aircraft wouldn't let you near it for hire or work without appropriate training - whether you could legally do so or not
