So our lawn was 80% moss.. I bought a scarifier/rake (Einhell electric from the rainforest) and applied a liberal amount of 4-in-1 moss killed, lawn fertiliser about a week ago. This could be down for longer but given the amount I put down it did a great job quickly with nightly waterings to drive the chemicals home.
We have some shaded areas and some direct sun areas.
First highest level pass scarifying:

Look at that moss appearing!
Now.. after completing the first pass and manually raking together:

Boot and rake in there for size reference.. this is all dead moss from the first pass only..
Here's the scarifier (top) and rake (bottom) attachments:

Now the next step was to switch over the attachment to rake and give it a once over... lots more moss appeared...
Then switching back to the scarifier again for a second pass at 30deg from the last pass - even more moss..

Now you can see the brown patches of earth minus the moss...
So just having lunch, then second pass rake before aerating with a fork.. and then over seeding with sharp sand and lots of grass seed later on once the sun has dropped.. water... and keep watering each day..
We have some shaded areas and some direct sun areas.
First highest level pass scarifying:

Look at that moss appearing!
Now.. after completing the first pass and manually raking together:

Boot and rake in there for size reference.. this is all dead moss from the first pass only..
Here's the scarifier (top) and rake (bottom) attachments:

Now the next step was to switch over the attachment to rake and give it a once over... lots more moss appeared...
Then switching back to the scarifier again for a second pass at 30deg from the last pass - even more moss..

Now you can see the brown patches of earth minus the moss...
So just having lunch, then second pass rake before aerating with a fork.. and then over seeding with sharp sand and lots of grass seed later on once the sun has dropped.. water... and keep watering each day..