If you are *G flying SQ you have access to LH SEN lounges and UA RCC at FRA, plus one guest. If you are non *G flying SQ in first you also have access to LH SEN lounges and UA IFL at FRA, plus one guest. If you are non *G flying SQ in business you have access to LH business lounges and UA RCC, no guests.

All lounge types have showers (ie there is at least one LH SEN lounge and at least one LH business lounge with showers, as well as UA RCC/IFL), although some individual lounges do not.

For JFK-FRA-SIN the gate used is B high numbered gate (on a pier). On arrival head into the main part of terminal and up one level to SEN and business lounges with shower. UA lounge is further away in the C part of terminal. I wouldn't bother heading there - not enough time.

For SIN-FRA-JFK the gate used in in C part of the terminal (because of the extra security needed for US flight). Thus UA lounge is close and LH lounges slightly further away (but still close enough to use in the transit).

However, given the queues for showers at FRA I wouldn't expect to get one in the short transit (but would try nonetheless except if inbound flight is late).