Learn Russian: Здесь можно петь.

Unlike verbs such as хотеть (to want) which change depending on who is doing the action, можно is a special impersonal word. It means 'it is allowed' or 'it is possible.' Because there is no specific person doing the action (like 'I' or 'you' doing it), можно stays exactly the same.

Where is the subject (like 'he' or 'we') in this sentence?

There isn't one! In Russian, you can use можно plus an infinitive verb (like петь) to make a general statement meaning 'one may' or 'people are allowed to' do something. You don't need a word for 'one' or 'people' at all.

How is можно different from the verb мочь (can / to be able to)?

You use the conjugated verb мочь when a specific person has the physical ability to do something (for example, Я могу петь means 'I can sing'). You use the word можно to say that an action is permitted or generally possible for anyone, which is why it translates well to 'one may'.

Could I change the word order and say Можно петь здесь?

Yes, you could, but Здесь можно петь is the most natural way to say it. In Russian, the most important new information usually goes at the end of the sentence. Putting петь at the end emphasizes exactly what action is allowed in this place.