For fixed bridge models, there's still a wide range to choose from.
First, ask yourself what kind of tone you want. This will help you when it comes to choosing the body wood. Do you want a thick, warm sound like a Les Paul? In that case choose mahogany or something like it. Or do you prefer a cutting, bright lead tone? Soft woods like alder often produce the bright twangy sound like in Strats. If you're undecided, you could always go with a neutral wood like basswood and allow the pickups to shape the sound more.
Next, you gotta consider whether you'll be playing with distortion often or clean/mild overdrive. If you play a lot of metal, a guitar with active humbuckers like EMGs will interest you. Otherwise, you can always opt for aftermarket pickups to replace those on your guitar if you wish for a different sound after purchase.
For necks, there a lot of different sizes and widths, but generally Les Paul necks are one of the thicker ones and the thinnest are Ibanez Wizard necks, strat necks are somewhere in between. Also, consider the profile of the neck, do you want it to be a C shape, D shape, V shape? I would suggest that you try out the different types of necks in a store, as well as the difference between finished/painted and unfinished necks.
There are a lot of shops around in Singapore, look around the Pennisula area, Swee Lee. Try out the different types and see which one feels and sounds better to you, tone is rather subjective, so what we like may not be what you want.