Concerning the non-trivial zeros of the Riemann Zeta function, one can find quite a lot of literature on:
the rate of growth of the number of zeros along the vertical critical line,
the zero-free regions of the critical strip
bounds on the number of hypothetical non-trivial zeros inside the critical strip, but off the critical line (though none have been found as of today, and never will if the RH is true)
However, I was unable to find any work concerned with estimates of the maximum number of hypothetical non-trivial zeros which may possibly lie on a same horizontal section of the critical strip (i.e. just the horizontal lines for fixed t values). Of course, it is well known that the functional equation implies that any hypothetical zero (1/2-a+it) must be symmetrically mirrored on the other side of the critical line by a zero (1/2+a+it). I imagine that such a set of hypothetical zeros would need to be discrete, as zeros of holomorphic functions are isolated, but I have no hint whatsoever about its cardinality (instinctively, we might feel that at most there are probably finitely many).
But perhaps some of you reading this question might know better.