This is literally the first Google result when I search: How PC Fans Work
If a 3-wire fan doesn't work with 12V power then it's borken. Hooking up random wires until it works isn't likely to damage it.
A 4-wire fan which is expecting a PWM input on the 4th wire can be run at full speed by leaving that unconnected. So it also works with plain 12V power.
Brushed or brushless will both spin without power (unless the bearings are siezed) so that's not how you tell them apart.