The charism for this sort of thing is rare. It is still found in a few religious orders as a matter of course. But for people in the world – or for Religious in communties that do not practice physical penance – There is only one rule: Never, never, never undertake any physical penance, such as using a discipline (that’s what the little whip-thing is called), or wearing a hair shirt (I dare you to FIND one today), or any other instrument of penance, unless you have cleared it with a competent spiritual director. NEVER.
People are naturally curious about this stuff, but it is a curiosity that should be indulged at the practical level only when you have matured sufficiently in the spiritual life to discern the difference between curiosity and charism. That said, many of the saints whom we praise for their asceticism practiced these things in the zeal of their youth . . .
Simpler things that do not require ‘equipment’ may be undertaken in a token way in order to focus on some aspect of our Lord’s life. A priest I know, during his seminary days, used to sleep on the floor during Lent to unite himself with Christ who “had nowhere to lay his head.”
I know another person (layman) who is currently discerning with his director whether his sense of call to physical penance in reparation for his own sins and for the failings of Priests is authentic or a matter of personal pride . . .