The 386 was able to run Win95.
It was also the first to be able to address 4GB RAM - not that any one of us could afford that much RAM back then to test that feature
My last 386 was a DX-40 too and it ran the PC-Tools, Central Point and Norton desktops at various times.
I think I ran my first linux distro on my 486-DX though, but I am sure that others did run linux on a 386.
You'd have to go back a bit to find the right kernels though.
The latest modern distros have been ditching support for the old chipsets and archs acknowledging that the old distro versions are still available somewhere.
There's about 6 variations of DOS that will running on there like 4DOS,
although I think MSDOS 6.1.x probably won't and I can't remember why. Just an engram hanging around in my lobes.
You'll need to configure EMM386 to use the expanded memory and then RAMspeed (or similar) to cache the extended ram.

Oh memory lane >!!<