Almost anything can become a proper substrate of penance and mortification. Saying extra prayers is a sacrifice of time, especially when done on one's knees or in some other uncomfortable position. Deferring a meal, a drink of water, or a trip to the toilette will inflict a minor chastisement upon the flesh. Keeping your mouth shut when you would really like to say something, or acquiescing to someone else's will in an indifferent matter, all serve to detach one from self-love. There are a thousand opportunities each day to do little acts of penance like this.

Above that, you can fast, you can keep watch into the night, you can give alms.  I know that sleeping on the floor is popular with some people. You might want to at least consult a director before undertaking a serious physical trial like flagellation, but this too is an option.

As far as letting others know about what you are doing, the main thing is to not take any pride in your mortifications. Keep them hidden if it is possible to do so, but if it is not possible then just make sure you are not seeking attention with them and see to it that they do not disturb the lives of others.