Should I as an L2 dependent get a Social Security Number?

[Cdn spouse]

Does an L2 dependant from Canada benefit from an SSN if not working? And alternately is there any reason to avoid it.
It seems we could save taxes from income splitting but we will have to pay Canada the difference anyway, right? This will be our first year to file U.S. Taxes and it is clear as mud. Any tax advice for filing in both countries?
Thank You for your help.