Jill,
Tax equalisation is standard in most MNC's. Some do it to a greater or lesser extent (eg bonuses may or may not be tax equalised). It can be quite complicated but in its most extreme form usually works like this.
Say you earn 100 in Oz. Out of this you deduct a notional amount for housing, say 15, and tax, say 40, giving you a net 45. This amount gets cost of living adjusted to Singapore, say x1.5 = 67.5 (actual is higher). You will get this net per month. On top the company will/may pay housing, school fees, car allowance, medical care, flights home etc. All the tax on these is borne by the company.
Tax equalisation is easy but generally moving to Singapore the company notionally deducts more than you are actually paying. If you are in an industry with high bonuses then excluding tax equalisation on bonuses is worth doing.
As for double taxation, if you are moving here for > 1yr you will become non-resident in Oz and will not be liable for tax on your non-Oz sourced income, salary etc. You will be liable for tax on Oz sourced income, eg rents, but you can negatively gear and carry forward your losses for when you return. You can also satisfy your Australian tax liabilities on Australian dividends by simply having non-resident withholding tax (10%) deducted at source. Shares reinvested in a DRP do not attract NRWHT. Hope this helps