「責める」は、単に相手を非難することで、「咎める」は、ガミガミお説教したり、ストレスを発散させながら非難するイメージですね。

責める means to blame someone no mattar why he/she should be blamed. Even if the someone is righteous person, when he/she got to be attacked, we use 責める at any case. But 咎める means to blame someone who is out of moral. 咎める needs some sort of ethical reason.

I think 責める is used more often than 咎める. とがめる is not old word, however I have not heard this word so often in speaking. (I find this word in written words.) For expamle, we say '私を責めないで’(means 'Don't blame me.'), however '私をとがめないで’　souds unnatural　a bit.

In terms of subjective...（主観でいうと…） 責める　→　blame 咎める　→　Interrogate