Showing the language of TMs that halt on a decidable set of words is not in RE
Here is a reduction from coHALT (given a Turing machine, determine whether it doesn't halt on the empty input) to $L$. Given a Turing machine $M$, construct a new Turing machine $M'$ which acts as follows:


Run $M$ on the empty input.
If $M$ halts, interpret the input as a Turing machine and run in on the empty input.


If $M$ doesn't halt on the empty input, then the set of words on which $M'$ halts is the empty set, which is decidable, and so $M' \in L$. If $M$ does halt on the empty input, then the set of inputs on which $M'$ halts is HALT, which is not decidable, and so $M' \notin L$.