In 2009, Franck Cassez showed that the timed opacity problem, where an attacker can observe some actions with their timestamps and attempts to deduce information, is undecidable for timed automata (TAs). Moreover, he showed that the undecidability holds even for subclasses such as event-recording automata. In this article, we consider the same definition of opacity for several other subclasses of TAs: with restrictions on the number of clocks, of actions, on the nature of time, or on a new subclass called observable event-recording automata. We show that opacity can mostly be retrieved, except for one-action TAs and for one-clock TAs with epsilon-transitions, for which undecidability remains. We then exhibit a new decidable subclass in which the number of observations made by the attacker is limited.