Heisuke Hironaka's coming talk makes me wonder how the state of the work on that theme is. So far, I noticed (but didn't read) these papers:
Kawanoue, Hiraku, Toward resolution of singularities over a field of positive characteristic. I. Foundation; the language of the idealistic filtration, Publ. Res. Inst. Math. Sci. 43, No. 3, 819-909 (2007). ZBL1170.14012.
Edit: Another recent talk by Hironaka (in Vienna).