Is a particular mobile number connected to Telstra, Optus, Vodaphone, etc?
You can find out for free at this link.
Call direct to Voicemail / Messagebank for Optus mobiles.
In response to Tom's Comment (which is appreciated), I've conducted a check of an associate's mobile number who I know used to be on Telstra, but has changed to Optus. The photo below is the result of the search on his number. As you can see, it shows the "Original Network" as well as the "Current Network".
Another Mobile Carrier Look-up service is available at this link.