Would it be wise and economical to get one patent for a formula that has three totally different uses in three different categories ?
I mentioned this very briefly in my answer to your previous question, but from the sound of what you're asking, that's probably not only your best option, but it might be your only option.
A patent is on an invention, not an implementation or use-case for that invention.
A good way of thinking of that differentiation is if I designed a table. Now, of course, I use the word "table" here because we both know what that is, but the actual patent application would likely refer more to a flat surface with four legs.
What's worth understanding here, is that my patent here for "a table" really wouldn't say anything about how I use it. It would say "a flat surface with four legs," and it would then be up to manufacturers to make it useful for something.
If you came along and said "I'm going to use that flat surface and those four legs, but in my version, it's for people to sit on," that would not be patentable. You'd still be infringing on my patent if you tried to use it for something different.
Utility patents do require their invention to have a use (utility, in other words), but the patent is on the invention itself.
Bringing it back to your question about formulas, the same thing applies. You can't patent a formula's use any more than I could patent the table as a piece of furniture meant to hold objects. Your patent will be for a specific chemical formula (I imagine), then uses for that will be up to you and anyone who buys your product or licenses the patent.
It depends. If the categories are medical, then you would definitely get a separate patent for what is termed second medical use in some countries. For example, if you discover today that aspirin prevents hangnail, you could get a patent for that, even though aspirin is known.
Even in non-medical cases you could probably obtain multiple patents on a same formulation for different uses, with claims to different methods of use.
It is worth exploring why you would want more than one patent. Even if for patentability reasons you decide to file a single application, having separate patents can help you in licensing, providing (in the USA) that the claims of one granted patent are not obvious in view of the others. Buyers usually prefer to buy a patent outright rather than license. Another issue is that when you draft a patent for each field of use, it allows you to focus on the opportunities in each such field.
Say you discover a blue pigment, and it can be used for ink, as a catalyst for catalytic converters in cars and as a preservative in shampoo.
You can file a basic application for the formula and three more applications, one for each use. When granted, license out the original application per field of use and sell each patent to a user with deep pockets.
You may want to protect more than the formula. Even if the formula is identical for the uses, when actually used, the product may not be. For example, different uses may indicate a different carrier for the formula and each such combination could be patentable as well. Since the formula is probably not used in a vacuum but as part of some configuration, there may be IP in how using the formula requires/takes advantage of changes in the configuration.