There are no easy options for a shower in HKG. The lounge there does not have showers (they don't even have a bathroom). It's a nice lounge with a good selection of food and drinks, but we didn't spend much time there. The 2 hour layover goes quickly. You have to reclear security and that can take about 10-30 minutes depending on if you arrive with other flights. We used the rest of the time to just stretch our legs and do some quick duty free shopping. You start to re-board the flight about 40 minutes before departure, so in the end you really don't have to kill too much time. Your only option for a shower would be to use the United or Thai Airways lounge which are very long walks from where the Singapore flight departs.

The bad news about Singapore is that they do not allow you to use their lounge in Singapore unless you are departing on a SQ flight. Since you're arriving, you would not be able to use the Silver Kris Lounge on your 3 hour connection. I don't know if KLM has a lounge or not. But at this airport, there are other really good options. We flew SQ last year connecting to a Silk Air flight, so we could't use the SQ Lounge in SIN, so we used the Ambassador Lounge with our Priority Pass. It was very nice with a good selection of food and nice shower facilities.

On your departure from SIN, you'll have access to the very nice SilverKris Lounge and The Private Room. Be sure to tell your taxi driver you are departing in First Class and they will drop you at a private check in area. It's really nice. Porters come out and meet the cab and grab your luggage. You then have a private check in and an immigration officer that just handles the 1st class checkin area. Very nice and efficient. After immigration you are very close to the lounge.

Enjoy!