If you want to run Linux on STM32 Discovery board, more exactly STM32F429 Discovery board, it’s now possible thanks to Emcraft Systems’ BSP for STM32F429 Discovery Board. It will only work on that particular version of the discovery board as it features 64Mbit external SDRAM, which is required for uClinux, and missing on other STM32 Discovery boards.

You can either get u-boot and (uC)Linux from the company’s github account which contains the required patches, but lacks complete documentation, and a default config file for the kernel for example, or purchase a BSP ($33) with 3-month support, and pre-built u-boot and Linux binaries. There’s also publicly available documentation explaining how to flash the bootloader and the kernel among other things, and links to relevant external resources, without the need to spend any money.