How many Turing Machines are there that run in time $t$ or in space $s$ on inputs of length $k$?
Given $k$, consider the set of $\mathcal{M}_k = \{ M_i \mid i \in \mathbb{N} \}$ with $M_i$ defined by

if |x| = k
  accept
else
  simulate TM i on x


The set $\mathcal{M}_k$ is a subset of the Turing machines you want to count for all $s,t$. Since it is infinite, the set you want to count is infinite, too.

As far as I can tell, this result is stable against all restrictions you impose as long as you don't restrict the set of all feasible Turing machines to be finite.