How to choose the right ILS when the runway has more than one?
When an airport has multiple ILS approaches to the same runway (usually labelled ILS-Z, ILS-Y, ILS-X, ...), they all end up using the same ILS for that runway, but they differ in how you fly the approach until final.

Consider e.g. KTUP, which has an ILS-Y and ILS-Z approach to runway 36. Here is the chart for the ILS-Y:


(source)

This ILS approach starts at an RNAV waypoint GAGYE and therefore lists "RNAV 1-GSP required" as a requirement. The ILS-Z approach looks completely different:


(source)

This ILS approach has two IAFs: CEVSI on the OTB 12-DME arc and the VERON NDB. It therefore lists "ADF or DME required" as a requirement.

So, which one should you choose?


During flight planning, you should plan to use the approach that best fits your aircraft's equipment. If your aircraft does not have ADF or DME installed, ILS-Z in the example above would not be possible. On the other hand, if your aircraft is not RNAV approved, you cannot fly the ILS-Y. If you can do both, then file a route that ends up at whatever IAF works best for you.
During the flight, it should be clear which approach you should fly based on which waypoint you end up at. In the example above, if had filed a route to the VERON NDB, then you would fly the ILS-Z. If you filed a route to a beginning of a STAR, then you fly the matching approach from the last waypoint of the STAR (KTUP in my example does not have STARs, but other airports with multiple ILS approaches might). This would usually be the RNAV variant, since most STARs are RNAV anyway.
If still multiple approaches are possible, ATC should clear you for one in particular, e.g. via "Cessna 12345, cleared ILS Yankee approach runway 36".