No Arrivals Lounge at SIN, just various departure lounges as noted. However, since both arriving and departing passengers can mingle at this airport, if you have a same day ticket on SQ, then you can access one of these lounges. Access, however, depends upon your class of service on your onward SQ flight. If economy, you will have to use the STARGold lounges, which are not the same as the Silver Kris Business and First lounges in T2 or T3 for use only by SQ's own elites and passengers flying those classes. Also, you may arrive at T2, so that means the SATS lounge (and I am not sure it has showers...was there last week but didn't notice any).

As also noted, you may not have your bags checked onto the late SQ flight if on a separate ticket, so after the shower and going landside, you'll have to collect your luggage at customs. Since your flight is more than 6-hours later, you may not be able to recheck them until you return to the airport.

So a bit more information and we can give you more precise information.