I have a 100+ year old century home. And the floors are so squeaky it could wake the dead. Like all homes of its period the flooring was nailed and and naturally it squeaks loudly after the floors have undergone a century of expansion and contraction.

So I'm going to do the unthinkable: remove each old nail and replace each with a screw.

What are the best types of screws to use for this job?

Stainless steel or brass?

Philips head, or roberts?

Coarse or fine threaded?

Bear in mind that this house was built before the days of plywood and OSB so it goes: hardwood flooring > old timber subflooring > joists.

I don't even want to re-visit this job again and I'm willing to pay more for a screw that won't screw me. Also feel free to rip into me and tell me why this is a bad idea.

Are you working on the sub-floor? or a finished floor that goes over a sub-floor?
If you have an old floor with no sub-floor I suggest using a PowerNail 50-C to add top-nails which will blend in, through the surface instead of removing all the flooring and using screws instead of 8d nails. If there are loose areas where there are gaps or the flooring is not tight together it was probably cause by the flooring being a little damp when it was installed. The sub-floor is sometimes laid when damp because the use the planks after using them for concrete forms and whey haven't dried completely before being installed. They can shrink and become loose.
The 50-C will drive inch and a quarter cleats with a rubber mallet that will push the flooring down hard when the cleat is driven, setting the nail over a quarter inch below the surface. They hold really well. There is also a cleat top-nailer that drives 2 inch cleats The nail heads blend in with the grain if you aim for the dark park of the grain.

It's a finished floor that goes over a sub-floor. So while I do appreciate your very detailed suggestion it sounds like I'll need a different approach.

If you have access from below to see the loose areas you could glue a strip to the floor joist and the loose board using sub-floor adhesive or construction adhesive. A piece of half inch plywood or even a lath strip from plastering.Tack it up with a staple or finish nail till the glue sets. Don't walk on the area where it was glued for an hour or so if possible.

No bottom access unfortunately. Entire basement is newly finish and covered up with drywall from wall to ceiling.