Identifying an existing easement

I recently purchased an older property in Fremantle, Western Australia. The back yard has no direct access to the street, but the boundary has an old asbestos fence along it separating it from a driveway on the neighbouring property, and there is an old concrete pad in the back yard abutting the fence that looks like it may once have been the floor of a garage or carport. What is the easiest way to find out whether an old easement exists that might provide for access to the rear of my property via this driveway?

Not sure about WA but in Vic you can access all types of info inc easements, mortgage, owner details of land through landdata is available on line for a fee of around $30 will also provide any covenants applicable to land also.
landdata.com.au

hope this helps

Would I need to look at the records for my property, or the adjacent one whose driveway would give access to my back yard?

Being in WA, any easements, covenants, memorials etc need to be disclosed to the Buyer when you signed your Offer.

Did you use a Settlement Agent? They would also have a copy of the Title (from the previous owner, and then a new one with your name on it once it has been transferred... they should send you a copy of your one, I think), and they'd have any other documents attached such as easements, covenants, etc. Contact them and see if they have any.
Your Agent will also have had to purchase a Title to have the property listed in the first place.

Do you have a copy of the Title from the previous owners, or a copy of the new Title with your name on it?

Failing all that, you can purchase a copy of your title from landgate.wa.gov.au.