I can run $().SPServices.SPGetCurrentSite() from my root site on Office365 and get "/" which is the valid value.
Yes, there were problems in 2013.02, but I thought I had fixed them in 2014.01 (at least I believed I had).
I've spent some time stepping through $().SPServices.SPGetCurrentUser() and I see the problem. It's only going to occur in the root site, as lauzierj is seeing. It's in line 2372 of jquery.SPServices-2014.01.js.
url: thisWeb + "/_layouts/userdisp.aspx?Force=True&" + new Date().getTime()
The value of url will be something like this:
which is invalid.
Sorry for the trouble with this, lauzierj. A quick fix for you would be to replace line 2372 with this:
url: ((thisWeb === "/") ? "" : thisWeb) + "/_layouts/userdisp.aspx?Force=True&" + new Date().getTime(),
Note that you'd do this in the unminified version of SPServices.