In Singapore, can you apply for a credit card without a job or payslip?

If you want unsecured lending - will be very challenging...banks need some assurance they will be paid back after all

But if you want a credit card as a payment method - banks usually accept some security deposit (ie place with them 10k and you get a credit card with 10k limit) and that can get you want you want.
Another thing to consider if the 10k is a bit painful - the recent launches of the FinTech cards (eg Transferwise/Revolut if you can fund it somehow) can be an alternative. They are focused on FX but even if you aren't interested in that you can still use locally. You don't need to stake 10k but still need to prefund what what you expect to spend).

There is some pros and cons - check out my reviews on the respective pages for info