Generally, if the airline would let you choose a seat for free with a paid booking, you can do the same for an award booking (e.g. Qantas). A notable exception is BA, who will still charge you a lot of money!
If you’re booking through the same airline you intend to fly (e.g. Krisflyer points for an SQ flight), just make sure you note the record locator (usually a 6 digit mix of numbers and letters) when you book. This will allow you to manage your booking online on the relevant airline’s website and choose a seat.
If you’re booking with points from an airline that is different to the one you will fly (e.g. United points to fly on SQ) you will likely need to ring the airline with whom you booked and ask for the record locator (e.g. ring United to ask for SQ locator). This will then allow you to manage your booking through SQ’s website and assign a seat.
Some record locators work across multiple airlines, but some do not. For example, I recently booked a flight using AA miles on a mixture of Qantas and Qatar. AA gave me a record locator for their website, which also works with Qantas and allowed me to choose my seats. However, I needed to ring AA for my Qatar record locator to allow me to choose my Qatar seats.