Language of TMs such that one state is visited most often
After a deleted (wrong) answer I thought  about the following to prove that $L$ is RE for single tape Turing machines.

As you said in the question, the problem is "recognizing" $M \in L$ that doesn't halt.

Note that a non-halting TM $M \in L$ not necessarily must expand the tape; for example:

q,[ -&gt; q,[,R
q,a -&gt; q,b,R
q,] -&gt; q,],L
q,b -&gt; q,a,L
causes an endless ping-pong in the same state q
...[aaaaa]....
   ^head


Also note that a non-halting TM $M \in L$ can expand the tape without changing its state; for example:

q,[ -&gt; q,[,R
q,a -&gt; q,b,R
q,B -&gt; q,b,L
q,b -&gt; q,a,L
causes an endless expanding loop:
...[aaaaaBB...


But for both expanding and non-expanding cases, we must have that the Turing machine $M$ must continue forever in the same state otherwise it doesn't belong to $L$ (cardinality constraint).

But the halting problem for one-state one-tape Turing machine, on a blank tape except for a finite number of cells, is decidable. See for example Gabor T. Herman: The uniform halting problem for generalized one-state turing machines. Information and Control; Volume 15, Issue 4, October 1969, Pages 353-367):

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Introduction: The uniform halting problem (UH) can be stated as follows: Give a decision procedure which for any given TM will decide whether or not it has an immortal instantaneous description (ID). An ID is called immortal if it has no terminal successor.
... we assume that in an ID the tape must be blank except for some finite number of squares ...
In this paper we show the solvability of the UH for one-state TM's ...
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So, to enumerate the $M_i \in L$ start a dovetailed simulation of $M_1,M_2,M_3,....$.


if $M_i$ halts at steps $n$  then just count the states visited and outputs $M_i$ if there is a most visited state; 
otherwise, if $M_i$ doesn't halt at step $n$, and it is in state $q$; decide if the corresponding one-state TM halts or not. If it doesn't halt then $M$ belongs to $L$;...
... otherwise continue the dovetailed simulation (if it halts it will be detected at a later step)