I opened an account with HSBC no problem and I hadn't even received my employment pass at the time. I simply got a letter from my employer verifying my work status and earnings. You also need to provide official verification of your place of residence. I simply got a letter from the reception of my
Serviced Apartment stating I was staying there (note it must actually state the room number).
And wayhay a bank account with debit card and internet facilities.
Downside of this account is that you have to maintain a minimum amount in your account or you are penalised. I think its $2000 for savings account (this is a current account without a cheque book but it does have a debit card/atm card) or $5000 in a current account (you get a chequebook). Also downside with HSBC is lack of ATMS and hefty charges for using non HSBS/Maybank/Standard Chartered machines.
DBS insist upon a FIN Number i.e employment pass/work permit or a letter from a Singaporean DBS account holder supporting your application would also allow you to open an account. I have been recommended to open a DBS account as they seem to have the most ATMs around the place.
Hope this helps