@crispo - Bali is a beach / chill out focused so just do that. Not sure where you are staying but you might like to try a little diving / snorkeling as per advice above. If you can tour around a little or get away from the coast you can see a little of the more authentic Bali.
As for Singapore a few days is perfect to get a feel for the place. It really is pretty small so a combination of taxis and walking (assuming you get aclimatised to hummidty whilst in Bali)
Have a wander around the various "sub areas" China Town, Little India and Arab Street. Boat Quay is worth seeing if only to imagine how it must have been before all the tourist bars appeared. The Asian Civilisation Museum opposite is worth a look. Clarke Quay is too touristy for me.
I used to live on Club Street (close to China Town) so it's a personal favourite for bars and restaurants. If you like live music pop into Indochine for some good Asian food, cool atmosphere and ask "Pinto" to play you some guitar. Rooftop bar of Screening Room or Beaujolais on the corner are good.
Singapore classic dish is chilli crab (although I prefer black or white pepper) - you can get a great view of the nighttime skyline at No Sign Board Seafood at Esplanade and great seafood, other bars and restaurants there too. It's the "upmarket" version of the original in Geylang recommend by @stewartrc above.
If you like Indian food you might try some Roti Prata or the Fishhead Curry in Little India.
There is an area of restaurants/bars popular with expats calle Dempsey which is quite near the Botanical Gardens (which are worth a visit when it's not too hot).
Food is quite reasonable in Singapore by western standards, booze is expensive. If you want to try authentic grab a bite in one of the hawker centres / food courts (where the locals eat) will cost you just a few quid. General tip is to join the longest queue and have whatever they are having - the Maxwell food court near china town/club street never killed me.