Anonymous HIV testing does not require the person to provide any names, NRIC numbers or contact numbers. Instead, a number is usually assigned to the test which allows the person being tested to receive the test results. No personal particulars are recorded, even with positive results of the HIV tests.
There are quite a number of clinics in Singapore providing this service. You can just google for the list of clinics. You can type 'Anonymous testing for HIV', and the list of clinics will appear. You can also choose a clinic near your place. Take care and do remember to keep yourself safe and protected.