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In 2009, Franck Cassez showed that the timed opacity prob-1 lem, where an attacker can observe some actions with their timestamps 2 and attempts to deduce information, is undecidable for timed automata 3 (TAs). Moreover, he showed that the undecidability holds even for sub-4 classes such as event-recording automata. In this article, we consider the 5 same definition of opacity for several other subclasses of TAs: with re-6 strictions on the number of clocks, of actions, on the nature of time, or 7 on a new subclass called observable event-recording automata. We show 8 that opacity can mostly be retrieved, except for one-action TAs and for 9 one-clock TAs with e-transitions, for which undecidability remains. We 10 then exhibit a new decidable subclass in which the number of observa-11 tions made by the attacker is limited. 12