I was able to run Tinycore on a Second Gen Intel Galileo Board with Quark CPU.
I know the Galileo Board is x86 , but its a weird x86.
It was not really hard, just though about sharing the experience:https://github.com/edgardogho/GalileoTinycore
It is not much other than compiling the kernel and booting grub on the SD card. I had to modify inittab to that it uses /dev/ttyS1 instead of /dev/tty1 since this board has no video card.