Savings accounts
You can and should open an Australian bank account while you are still in Sg. I personally chose NAB because it offered the best saving interest rates at that time among the various banks that allow migrants to open accounts prior to landing in Australia. NAB doesn't charge account fees or ATM fees and doesn't impose a minimum balance. NAB is one of the big 4 banks in Australia (Westpac, Commonwealth and ANZ being the other 3).
If you use NAB, open a Classic account and an iSaver account. The Classic account is your so-called transaction account which is tied to your debit card and your cheque book. It earns practically zero interest (currently 0.01%). The isaver account is your saving account. It earns about 3% for the first 4 months. Your transaction account is your working account so you have to ensure there is enough balance for your ATM withdrawals and purchases. You need to transfer money from your savings to the transaction account via online. It sucks that you cannot transfer at an ATM. Luckily almost everyone has a smartphone nowadays. You can do that through their app.
Once you arrive in Australia, go to the specified branch and validate your identity by presenting yourself and your passport. You cannot touch your money till this is done. You can transfer AUD in but you cannot move it out. Also arrange for a debit card which also acts as an ATM card. A debit card allows you to pay for things without the need of carrying too much cash and for purchasing stuff online. The debit card is extremely useful until you get your first oz credit card. Whether you should get and the pros & cons of using a credit card is another story.
You might want to open other bank accounts when you are more settled in Australia. Deposits in each bank are guaranteed by the Australian government for up to $250k. So if you happen to have more than $250k, you can reduce your "risk" by splitting your money into different banks. If you are kiasu like me, always check if the bank you are going to transact with is in the Oz government's list of eligible institutions. If you Google interest rates, you may see some unfamiliar names promising much higher interest.
Also several banks (including NAB) preferential interest rates are good only for the initial few months. Online banks offer better savings interest. The catch is : these online banks won't have a branch if &/ when you need face-to-face interaction. The online banks' savings interest are usually even better than most term deposits (or what Sporeans called fixed deposits). Currently the best 12month term deposit interest rate is 2.80% while ME Bank Online Savings (OSA) offers 2.95%. The OSA offers the additional benefits of liquidity (no-lockin) and monthly rest interest (though negligible).
You can read more about the government guarantees here:
Government guarantees on bank deposits
For a quick comparison of interest rates:
Canstar deposit comparisons
Infochoice term deposit interests comparisons
ME Bank
Members Equity Bank (or ME Bank in short) is based in Melbourne, VIC. But you can open an account anywhere in Australia. At the moment its OSA offers 2.95%. The catch is you must tap their debit card at least once a week. Their debit card comes in several cool colours like orange. A couple of years back, they even offered $50 (or was it $100?) just to open an account with them. You can withdraw cash at any Westpac ATM without any charge.
ME Bank Online Savings Account
UBank
UBank is the part of NAB and it is an online bank. No, unfortunately you cannot perform bank transactions at a NAB branch's counter. Their best savings interest is 2.87% which is probably comparable to 12month term deposits but with the advantage of liquidity of a saving accounts. With UBank, like NAB, you can use any Redi ATM without any charge.
UBank savings accounts
I can't claim whatever I wrote so far is complete - it's likely not. There maybe some institutions out there offering a better deal. Do your research. I know some banks offer higher interest but you will have to do regular (monthly) deposits. Eg ING offers 2.80% on savings but you will have to deposit at least $1000 monthly. Is this easier than tapping the ME Bank debit card once a week? I guess it is up to one's preference.
More information on savings accounts
Once you have a tax file number (TFN) with ATO, remember to inform the bank. If the bank doesn't have info on your tax residency, the bank will withhold part of your interest earned which you can claim back when you file your tax (lay chay). When the bank has your TFN, you will also notice the interest earned with the bank is auto-filled in when you lodge your tax online.
Lastly, in case you are wondering how the wonderful interest you are earning in Oz measured up to Oz inflation, inflation information can be found at RBA websites. RBA offers a calculator that allows you look at inflation numbers between 1966 to 2016. Using the calculator, $100 in 2015 is as good as $101.28 in 2016 giving an annual inflation of 1.28%.
RBA inflation calculator