Useful Everywhere
Known as Universal Cure No. 1, and for the more enterprising ones - Universal Currency. Black Tape is used liberally in the SAF to make things stay in place or look neat.
Toilet Roll
Universal Currency No. 2. Even though this is provided in some places, you always end up needing them exactly when they run out. Additionally, you'll need your own for field camp (however, most people hold their bowels).
Spectacle bands/hooks and spare spectacles
Unless you are very careful.
Books
The old observation that in the Army, you wait to rush, and rush to wait is true, as you will realize quickly. Though BMT can be quite stressful with tight timings, there will always be downtime (especially if you are from a PTP batch). As such, books can be a good way to pass the time, as well as keep the brain working.
Stationery
There is quite a lot of studying to be done, if you are conscientious about it.
Bunk Essentials
Alarm Clock
Not entirely essential, as most bunks will have ten alarm clocks ringing in the morning already. However, it's good to not rely on anyone else.
Hangers, Pegs
Self-explanatory. It is possible to keep a laundry bag and just wash clothes over the weekends, but you definitely should bring some for the initial confinement and also if you are awarded any extra ones.
Coins
Different companies will have different policies, but most will allow recruits to purchase drinks from the vending machine at some point in time. Depending on how often/when you are allowed to, you may want to bring several tens of dollars' worth of coins. From personal observation, PTP recruits generally seem to meet more restrictions on this topic.
Extra Attires
If you can spare the money, buying extra attires is a real convenience that lets you delay laundry sessions. Otherwise, when you run out of clothes, you will have to wake up extra early (and normally is already quite early!).
Washing Powder
For aforementioned laundry. It's not unheard of to just use water, however, as SAF clothes are surprisingly good at staying stink-free.
Toiletries
This is actually provided, but you'll want to bring one full extra set for two reasons: one, it's more comfortable and familiar; two, you don't have to keep rearranging the provided set for stand-by-beds.
Underwear
Of course lah.
Slippers
The provided sandals are quite comfortable but strangely heavy. Also, wearing sandals can be quite leceh.
Nail Clipper
Bearings check includes nails, so don't forget!
Dehumidifier & Air Freshener
These go hand in hand in making your bunk a nicer place to live in. One of the key advantages this brings is that, subconsciously, people feel that a nice-smelling place is a cleaner one: commanders will give you more slack during inspections.
Febreeze
I personally never bothered with this, but this is the wash-clothes-without-washing-clothes secret. Just spray your dirty clothes with this and tadah!
Rags
Area cleaning is a daily event in the everyday life of a recruit, so bring these to make your life easier. Your rags are likely to get passed here and there so make sure you bring extras because some will definitely get lost.
Make Life Easier
Facial Cleanser
You will definitely want this, with beads if possible, once you start using camo - especially when commanders come around checking.
Lozenges
Tekong cough is a really unique disease that happens to many people - when I was a recruit, I thought that it was just a part of army life due to all the shouting and singing; until I POPed and realized that even though I was still shouting and singing, I stopped coughing. I guess it's really Tekong!
Blister Patch
Your civilian feet will not be able to take the stress of route marches, so get some of these for the tougher ones. Note that these tend to be on the pricey side of things, so it's not a confirm-must-get.
High Quality Socks and Anti-Abrasion cream
The e-Mart does sell higher quality socks than the standard issue ones, but these cost money. Useful to have at least a set for route marches. Anti-Abrasion cream is obvious.
Pain Relief
Don't need to explain this one.
Scotchbrite
This is illegal in some companies, but if your sergeants haven't said anything about them, they are very useful for rifle cleaning (cuts literally ~60% of your work).
Others
Non-camera Handphone
You will not be allowed to charge your phone with a wall socket, so bring either a battery operated charger, spare battery, or spare phone.
Swimming Goggles
No rush, you will be informed when you actually need this, but at some point you will. PTP recruits will use the pool a lot more than Enhanced ones, so be double sure to get these if you are one.
Black Rubber Watch
Almost compulsory, and useful to keep a separate time since company clocks tend to be quite different (i.e. much faster) than civilian ones.