My scooter battery can not self start the scooter any more. Can I use the battery for any other work?

Do you mean you want to take the battery out of your scooter and use it for a different purpose?
Like run a fan in your garage?
Most electrical appliances run off of AC current.
So the battery is useless in that regard.
You can use it to run a flash light, but it's impractical.
Don't waste your time trying to get some use out of it.
Bring it to a battery recycler (bike/car shop, auto parts store).

Any battery of the same voltage and with enough cranking amperage will work. The starter will only draw the amperage it needs, so don't believe any urban legends about needing small batteries or tiny jumper cables. Just be sure it is connected with proper polarity [+ to + & - to -]

Former GNB [battery Co.] Rep.

Try the Wally-World 12AH 12 volt battery, that is what I'm running in my Chinese scoot.