When you arrive HKG you have less than an hour before boarding to SIN, so make a beeline to the United Club and ask to get on the waiting list for a shower when you check in at the front desk. They will take your boarding pass and page you when a shower is ready. In the meantime you can sample their buffet. As departure time nears people will be giving up and getting their boarding passes back - you may have time to take a quick shower. Don't forget to get your boarding pass back when you finish.
Upon arrival at SIN follow the crowd to the end of the B corridor, but do not follow them left downstairs to immigration and baggage claim. If you want a SIN stamp in your passport as a souvenir, make it last on your list - otherwise you will not be able to get back in.
On the right at the end of the B corridor, will be a lot of arm-less seats with many people sleeping. Across on the left will be a closed info counter next to wifi podiums and the transfer station, including the United counter which opens at 3:30 AM, sometimes earlier. Check in even if you already have your boarding passes. As gold, they should give you a complimentary pass to the SATS lounge. Give it a try. When checking in at the SATS Lounge front desk, ask for a towel for a shower. They have a nice buffet.
You are free to mill around, but be prepared to display proof of your next connection to roving security (a plainclothes officer accompanied by armed soldiers). A copy of your itinerary or your onward boarding passes will do.
Any open info desk will give you a free wifi pass-code upon displaying your passport. Upstairs on the right is a 24 hr food court with a Burger King. Also upstairs is a free sleeping lounge which is usually full.
The Ambassador transit hotel upstairs has rates in 6 hour blocks. With deplaning and boarding - you will be at SIN about 3 hours.
About the sdc - most first/business seats are still open which may translate into economy+ aisle seats opening up. If you are up to it, you can check availability 1:05 AM Dec 7, the earliest time to book a sdc for your itinerary. For a return sdc, call UA from SIN. No one at SIN can help you. Also many UA agents are grossly misinformed - just call until you get a competent one.
On your return trip, your HKG layover is 3 hours - plenty of time for your third shower and for exploring other lounges.
Upon return when immigration questions you, I would advise being honest about being on a mileage run. I have had no problems so far. I would advise pulling a puffed up roller carry-on behind you through customs. Almost every time I travel very light on a long haul, I am searched and questioned - never so far with a roller carry-on.
Many of my UA trips to SIN are as springboards to places such as BKK, SGN, MEL, PER, CMB, GOI on discount airlines- a far cheaper alternative.
Bon Voyage!