Linear Bounded Automaton that accepts all strings
For Turing machines another problem is also undecidable: whether it accepts any string at all. This is already undecidable for LBA. Basically the proof is by oberving that we can replace a TM accepting string $w$ by a LBA accepting the `trace of the computation' by the TM on $w$. The verification of the trace of the computation does not need more space than the computation itself, and thus can be verified by a LBA.

The question whether a device accepts all strings is already undecidable on a lower level of the Chomsky hierarchy, the context-free languages (CF grammars,or pushdown automata).