Short answer to your specific question - an available and fully functioning ILS will provide you with two "legal" options, a Precision Approach and a Nonprecision approach (flown LOC only.) That would leave RNAV as an acceptable third type of approach.
Appendix 7 can be found on page A-15 and A-16 of the Instrument Rating ACS (currently FAA-S-ACS-8B).
This lists the requirements for the Nonprecision and Precision Approach Tasks in Area of Operation VI., Instrument Approach Procedures.
Task A. Nonprecision Approach
The evaluator will select nonprecision approaches representative of the type that the applicant is likely to use. The choices must use at least two different types of navigational aids.
Examples of acceptable nonprecision approaches include: VOR, VOR/DME, LOC procedures on an ILS, LDA, RNAV (RNP) or RNAV (GPS) to LNAV, LNAV/VNAV or LPV line of minima as long as the LPV DA is greater than 300 feet HAT. The equipment must be installed and the database must be current and qualified to fly GPS-based approaches.
"The applicant must accomplish at least two nonprecision approaches in simulated or actual weather conditions.
• One must include a procedure turn or, in the case of a GPS-based approach, a Terminal Arrival Area (TAA) procedure.
• At least one must be flown without the use of autopilot and without the assistance of radar vectors. The yaw damper and flight director are not considered parts of the autopilot for purposes of this Task.
• One is expected to be flown with reference to backup or partial panel instrumentation or navigation display, depending on the aircraft’s instrument avionics configuration, representing the failure mode(s) most realistic for the equipment used."
The evaluator has discretion to have the applicant perform a landing or a missed approach at the completion of each non precision approach.
Task B. Precision Approach
The applicant must accomplish a precision approach to the decision altitude (DA) using aircraft navigational equipment for centerline and vertical guidance in simulated or actual instrument conditions. Acceptable instrument approaches for this part of the practical test are the ILS and GLS. In addition, if the installed equipment and database is current and qualified for IFR flight and approaches to LPV minima, an LPV minima approach can be flown to demonstrate precision approach proficiency if the LPV DA is equal to or less than 300 feet HAT.
The evaluator has discretion to have the applicant perform a landing or a missed approach at the completion of the precision approach.
Task VI.D., Circling Approach, must also be accomplished somewhere within this matrix.
I attempt to meet these requirements with three approaches. It's not always possible based on ATC and traffic flow. We're mixing quite a lot of requirements from Appendix 7 into the two Nonprecision approach tasks.
They remain an ACS task (Task V.A., Intercepting and Tracking Navigational Systems and Arcs) and are still tested under the Instrument Rating Airplane ACS. But, they are only tested at the evaluator's discretion and under the following conditions:
"While the applicant is expected to be able to fly DME Arcs, they may be selected for testing only if they are charted and available."
This makes it clear to the applicant and instructor that training must be accomplished for DME arcs. Also, as an ACS task, at least one knowledge element must be evaluated. Both of the knowledge elements in that task reference arcs.
The applicant could "skip" training DME arcs and perhaps get lucky on the checkride, but the very unlucky possible result is that the evaluator will select a charted and available for arc for testing. It would be unlikely the outcome would be a good one for the applicant in that case.
Hope this helps,
. thanks!