Ever determined, next I'll try doing a "HDD install" of core.gz's contents to a CF card and try booting it in an IDE adapter. Not the Tiny Core way, but it should do for testing whether the binaries run on a 486 at least (and could run the "proper" way on one with more RAM).
I finally got one of my 486 laptops to boot TC12 (with the i486-rebuilt rootfs.gz from earlier in the thread). Plenty of problems with running out of RAM (16MB in that case, using a laptop with an easily accessible HDD bay), but no "illegal instruction" type errors. The exact CPU model is a 486DX4 100MHz. So TC12 now gets my tick of 486-compatibility.
The details are that I did a HDD install (on a real HDD because the CF adapter proved physically difficult to accomodate), along with a 60MB swap partition. I still had lots of trouble with running out of RAM while the tc-config script was running at boot, so I ended up making lots of modifications to that script. The most important changes were probably commenting out the line "/sbin/udevadm trigger --action=add 2>&1 >/dev/null &" to disable "hotplug" support, and moving the enabling of swap partitions directly after fstab creation (compressed swap in RAM was disabled with the nozswap bootcode). Starting udevadm doesn't really work even after swap is enabled (though with swap it doesn't crash the whole system at least).
Once booted, "free" shows only 6MB of RAM available, so I guess the Linux kernel must claim 10MB for itself. It's possible to boot without the swap space enabled and use the shell, but swap is required if you load extensions otherwise it runs out of RAM when loading the required kernel modules (I commented out the pre-loading of the "loop" and "squashfs" modules in tc-config).
If it had around 80MB of RAM (I believe some 486 motherboards allowed for this, probably designed for server applications) it might be practical for some tasks, as RAM seems to be much more of a limit than the CPU (this being among the fastest models of the 486).
PS. @gadget42: I'm posting this from DSL on a Pentium 1 (Dillo web browser) - it ain't all that bad!