Most employers have asked me to confirm my last salary at some point in the interview process, usually in the last HR interview before they look at salary negotiations. This has been true in most countries i've worked in. But Singapore is the only place where employers have asked me (consistently) to produce paystubs and/or letter confirming last drawn salary after negotiations have been completed. I've never lied, so I have no idea what would happen if my paystub didn't match my claims.
Most HR departments have some benchmark that they use as a basis for salary increments. This benchmark varies a bit based on the skills, the position and how hot the job market is. They will also have a budget for the role, which is the maximum amount of money they can pay for the role. If your through the interview process and the hiring manager wants you, you can usually negotiate up to or a little past that, somewhere between HR's benchmark and the budget they have for the role. If your current salary is around the budget for the role (i.e. they think you are already at the top of the salary range for your skills/position) then you can expect things to stay flat; there is no guaranteed increment just because your hopping to a new job.
What you need to be most concerned about is your salary expectations and not your current salary. Presumably this is expectation is based on some rationale that you have the ability to articulate. During salary negotiations, the ability to articulate this well and convince the HR manager, plus how much the hiring manager wants you will dictate whether your able to meet the HR benchmark or exceed it (or walk away with an offer for your current salary).
That being said, I have ruled out some candidates based on ridiculous salary expectations. This has usually been when the expectations have exceeded my budget for a position. But I've never rejected a candidate just because their is a huge gap in their current salary vs. their salary expectation (as long as it's in budget).
There are a number of candidates who didn't get their expected salary increase when we've finally made/negotiated an offer because it was too far from their current salary. But they certainly didn't get dis-qualified from the interview.
I should probably point out my experience is mostly from a large MNC point of view, some of it in SG. But all in technology. Singapore based companies / SMEs may operate a bit differently.
I am happily working for them and do my best regardless of work stress or whatsoever !. 