SIN - it's very well organized here. I'd recommend the transit tour offered by SIA if you want the most bang for the buck.
If you want to shop - then I'd head to the intersection of Scotts Rd and Orchard Road (Singapore Marriott anchors this corner) and spend several hours shopping. Deals here are not what they were in the early 1980's.
I'd leave 45 minutes transit if you plan to use the MRT (subway) to return to the airport from Orchard Road to be on the safe side.
FRA - you get into town here, you need to take the train in the basement of the airport. They run every 10-15 minutes I believe depending on the time of day. Not known for shopping and I didn't see many great food options. Mostly scenery here.
You arrive at the train station which is about 10 minutes walk into town.
When I was there on a sunday, there was not much happening in town. Most stores were closed, but nice to see some European architecture and Platz's.