Over the years, people have conducted animal studies trying to shed light on this matter. Some, but not all, studies have shown a few of the chemical compounds in hair dyes to be teratogenic (causing birth defects). However, in many cases the animals were exposed to extremely high doses of these chemicals, doses in no way equivalent to the exposure of the woman who colours her hair every month or two.

Clearly, the chemicals in both permanent and semi-permanent hair dyes are not highly toxic. These chemicals have been around a long time, and no research shows they cause defects in newborns, so colouring your hair during pregnancy is probably safe. Plus, I apply the dyes safely (using gloves in a well-ventilated room, and not leaving solutions on for excessive periods of time), you don't really absorb much of the chemicals into your system.