I'm posting this mostly because I found it very hard to find information about walk-ins, and to update the forum with yet another data point. I'm aware these things are a little fuzzy, YMMV, etc.
TL;DR: ICA no longer accepts walk-ins. So, check for cancellations the day before if you can't wait the normal 6+ months waiting period.
I missed my first PR appointment (actually, cancelled it) after my university took 4 weeks to send me paperwork that was guaranteed 3 day with DHL (lesson 2: have all your paperwork ready BEFORE you book). I think they only set about doing it when I emailed them asking what was going on... anyway.
So, I decided to try the "cancellation" e-appointment method, but there was nothing on yesterday for today. Some people mentioned there was a walk-in queue (e.g. http://news.asiaone.com/News/the+Strait ... 18499.html). So, I thought why not try it.
Turned up at 7:10am at ICA this Thursday morning. The queue outside the building only had about 20 people in it. At around 7:40am, the queue really started growing and the security guards asked people to tighten up. I couldn't see how far it went, but it was definitely over 100 people. So, turn up before 7:40am. A woman tried arguing to queue jump, alas, the guard made it clear that there would be no exceptions.
At 8 on the dot, they opened the gate and the queue marched in a fast and orderly fashion towards the gate. Quite surprising not to see any more kiasu behaviour although that came later when coming back out of the lifts. Amazingly, I was second (and last!) in line after catching the first lift up, most people chose the escalators which are slower. The counter staff was friendly but effectively said "no walk-ins since 2009, you gotta do e-appointment, no appointments until next year". Well, that explains the lack of queue. Took a walk round the block, decided to go back and ask if they had any last minute cancellations.
Another lady told me this time to fill in a form, take a ticket, and she'd come see me. I waited about half an hour, went to the counter, where another lady told me the government was trying to "discourage walk-ins" and so "didn't accept them" but she'd talk to her superior. The first lady saw me about 10 minutes later, and said effectively the same thing, but booked me a cancelled slot in a few weeks. She offered to check all my documents, which she was kind enough to do.
So, don't walk-in.
Fun fact: one of the counter staff (can't remember which) asked me to put something else for "Race", apparently "Caucasian" is too broad and I should put my country of origin.