As others have said, no airline (not even Singapore Airlines) has an arrivals lounge at SIN. This is mainly because the people normally using an arrivals lounge tend to have other options, such as:

1. Connecting passengers get lounge access in the departures area. BA and SQ both have showers in their lounges, and there are also options such as Priority Pass and even lounges you can pay to use. There is also the pool on the roof of T1.

2. Passengers arriving at SIN can be in their hotel (or home!) 20-30 minutes after leaving the airport - there's rarely much of a wait for a taxi. Hotels are pretty good about early check-in and allowing people access to showers etc if needed.