If the flights were purchased as a connection then yes, SQ does take responsibility for the connection. However, what you are asking is what would happen if you purchased the two flights individually and not as connecting flights.

As far as SQ is concerned you do not have a connection, and they will not consolidate the bookings into one as this will cause a violation of the fare rules for the PER-SIN-PER and SIN-DXB-SIN bookings. By consolidating the bookings, it would change it to a PER-DXB-PER one and that has it's own fare basis. You can ask them to 'link' the two bookings if that's something they still do so that you can check your bags through to DXB at PER (and vice versa), but if they don't do this anymore then you will need to do so a the transit desk in SIN.

Note that this also means that in the system you have 2 separate bookings and not 1 booking and while there is a high chance SQ will re-book you in case of IRROP, they are certainly not obligated to do so.