In general there are two schemes that allow you to rent a flat from HDB at subsidised rates: Public Rental Scheme and Parenthood Provisional Housing Scheme.
The Public Rental Scheme is heavily subsided and caters to Singaporean households with no other housing options. Since the supply of rental flats is limited, applicants must meet the eligibility criteria set by HDB:
Parenthood Provisional Housing Scheme on the other hand, is meant to provide temporary housing for families as they await completion of their HDB flats. You can read in detail about the eligibility criteria on the HDB website.
If you do not qualify for either of these schemes, the alternative would be to rent from the open market.