I cannot be sure, but I've heard enough anecdotes to have an inkling that many people still do not believe their vote is secret.
OK. Actually, I have met some people who absolutely believe that their vote is not secret.
They have their reasons.
Firstly, they notice that their ballot paper has a serial number. This makes every ballot traceable. If they don't intend to trace how one votes, then why is there a serial number, right? [play ominous music!]
Secondly, the election officials at the polling station reads out the voters name and some numbers. And if you are paying attention, it is to those people in white sitting opposite them. People in WHITE! [play MORE ominous music!]
Thirdly, the election official WRITES down something (I don't know - name, NRIC number, etc) on the counterfoil of the ballot! [Play EVEN MORE OMINOUS music!]
Well, I have voted at a few elections by now, watching how they process this voter, and I can confirm that something of the sort happens.
But, the above is a little vague, and because it is a little vague, it is subject to interpretation and some interpretations can be rather ominous.
First of all, the people in WHITE are, as you suspect, PAP members. Probably. Or at least PAP volunteers. At some polling stations, you MAY see another person seated next to them. These (those in white and those others) are the Polling Agents (PA) of the candidates. These PA are at the polling station to observe that voting is carried out fairly and transparently, and in accordance with the law.
The reason why you see mainly people in White is because the PAP has the manpower and the volunteers to roster a full team in two shifts (at least) to attend to the polling for the full 12 hours (from 8 am to 8 pm). I suspect that by now the WP may have enough volunteers to do so, but I cannot be sure. I don't vote in a WP GRC or SMC (or one challenged by WP) so I have not observed firsthand.
But most other opposition are so small and so under-resourced that they do not have the volunteers or manpower to attend to EVERY polling station and cover EVERY ballot box.
If a GRC has say 200 ballot boxes that means a candidate or party will need 400 PAs in 2 shifts (6 hours each). That's 400 volunteers to be PA just to cover 1 GRC. Contest in 2 GRCs and your requirement may well double. And 200 ballot boxes are just an estimate. I do not know if this is too many or too few. For larger GRCs, there may well be more ballot boxes.
Serial numbers of Ballot Paper
Ballot papers are serialised for control purposes, to prevent "ballot box stuffing" and other fraud where ballots are duplicated. It is for the same reason that SGD notes are serialised, while so-called "Banana notes" (from the Japanese Occupation in Singapore) are not, and are effectively worthless.
If ballots are not controlled (via serial numbers) then it would be easy to duplicate ballot blanks and fraudulent votes can be cast. And the principle of "one person one vote" can be violated.
Serialised ballot paper ensures that every registered voter is given one vote. To record the recipient of the ballot paper, the serial number of the voter in the register of electors (NOT the NRIC number) is recorded on the counterfoil of the ballot paper.
See para 5.5 of this guide from the elections dept:
So? This means my vote can be traced and it is not secret, lah?
Not exactly true.
From the description of the process it sounds like a 3-key system.
The ballot paper has a serial number because it is (de facto or actually) a "controlled document". That means if you happen to find a single ballot paper, you can tell who the vote is for (if it is properly marked with an "X" in the box of candidate), but you do not know the identity of the voter.
But there is a serial number.
Yes, so say you have the ballot paper, and the counterfoil that matches that ballot paper's serial number. You flip to the matching counterfoil, and you find that on the counterfoil there is a 4-digit number. It is the serial number of the voter on the Register of Electors!
AHA! so we know who is the voter of that ballot paper!
Only if you also have the register of electors for that polling station.
So to summarise,
1) The ballot paper has a running serial number. Based on that serial number you can find out... nothing about the voter.
2) The counterfoil of the ballot paper with the matching serial number of the ballot paper will have the serial number of the voter from the Register of Electors. So with the serial number of the ballot paper, and the matching counterfoil, you will get the serial number of the voter from the Register of Electors. And NOW you can find out.... still nothing about the voter.
3) The Register of Electors has the serial number of the voter, and NOW, you can traced the serial number written on the counterfoil to the person who had been given that specific ballot paper, and if you can look at the ballot paper, you will know how the person voted!
So... Voting is NOT secret.
Well... actually it still is, but it just means that there is a process to confirm that a specific ballot paper is legitimate and authentic, and had been properly issued to a genuine, registered voter.
This is a three-key system which means that if you have just one key, either the ballot paper, the counterfoil or the register of electors, you would be unable to tell how a specific person voted.
Even if you have two of the keys - say the ballot paper and the counterfoil, all you will have is the serial number of the voter on the register. You would not know who the voter is. If you have the ballot paper and the register, you would not be able to link any voter to the ballot paper. So the vote is still secret.
If you have the counterfoil and the register you can identify the ballot paper that was issued to a specific voter. But without that specific ballot paper, you would not know how the voters voted.
It is only if you have access to ALL three "keys" - the register, the counterfoil, and the huge pile of ballot papers - and the luxury of time to go through the whole pile of ballot papers, trace every ballot paper to their counterfoil and then to the voter issued with the counterfoil that you can discover who cast a specific vote.
If you intend to find out a specific individual's vote you will have to trace the name of the voter on the register (it is not in alphabetical order if what I could glimpse was correct), find out the serial number of that voter, then look through the counterfoil to find that serial number so you will know what is the serial number of the ballot paper that was issued to the voter, and then sieve through the pile of ballot papers to find that one ballot to find out how that person voted.
If you have the poll card of the voter, which has the serial number, you will be saved the first step. But you would still have to find out from the counterfoil, the serial number of the ballot paper, and then you would still have to find that specific ballot paper.
So yes, it is POSSIBLE to trace every vote, but it is not easy.
And so for all practical purpose, your vote is secret.
But, it's okay. Next election, people will still believe that their vote is not secret.
It is good to be critical, and sceptical, but at some point, one slips into "intractable" and "incorrigible" territory. And we have better things to do than to keep trying to convince the paranoid that their vote is secret.