We found it really easy to open a current account - we use UOB (United Overseas Bank) which a fellow Brit in our company recommended. We've been very satisfied with their service, very speedy at sending transfers of $$ back to the UK and the US and as it is a major bank in Sing, (one of the big 4, just like those in the UK) there are branches and ATM's all over the island. We actually completed the paperwork to open the a/c at their London branch, just a few hundred yards from Liverpool St station - we took our passports to this branch, signed some papers and they were forwarded to their main branch in Boat Quay (Raffles Place) in Singapore. It was great as the morning after we arrived, we turned up at the main branch and collected our new cheque book and NETS card (to withdraw cash from ATM's and for paying for goods or supermarket shopping - as it is also a debit card). We were given an allowance upon arrival so it was great to be able to have the a/c up and running immediately rather than carry large amounts of cash around, which we hate doing. We had to provide a copy of the job offer to confirm that we were being given an expat posting in Singapore, and were able to use the company address to open the account.
We also applied for a UOB credit card (visa) but generally prefer using Amex for airmiles and for faster replacement/aid in emergencies. I have to say that their website is not too great (http://www.uob.com.sg) but just look under personal banking and current accounts. It is a load of rubbish the part saying to open an a/c that you need to be a Singapore citizen or permanent resident as they have thousands of expat account holders, I think it's an oversight on their website. However, it lists the address and phone no. of their London office (which is really a corporate branch) but they will be able to advise on how to go about setting up a current account.
Hope this helps and Happy Christmas!