Permanent hair dye and straighteners may increase breast cancer risk. Scientists found that women who use permanent hair dye and chemical hair straighteners have a higher risk of developing breast cancer than women who don't use these products. The study suggests that breast cancer risk increased with more frequent use of these chemical hair products. Source 9m.
I believe this. I know it makes ladies hair thin and fall out which isn't normal, and I believe the two are related.
Personal use of permanent hair dye not associated with greater risk of most cancers or cancer death. Women who use permanent hair dye products to colour their hair at home do not experience greater risk of most cancers or greater cancer related mortality. Source 6x.