The Moscow subway system was considered for use as bomb shelters as early as World War II (against conventional weapons). They were also extensively studied for use as nuclear blast shelters in the Cold War. It should be noted that by the Soviet's own estimations, the subways were not large enough to shelter the entire Moscow population; they could accommodate maybe 530,000 or so (out of a metro area of several millions), assuming you had adequate warning to move people down there.

In the USA, I don't know of any subway metro systems that were specifically designed for this task, though the vaulted ceilings of the central Washington Metro system have been noted in contemporary discussion of public shelters as being pretty good for blast pressure support. (The New York City subway system, by contrast, was clearly not specifically developed with this in mind — it is much older on the whole — but any underground shelter is probably better than any above-ground shelter, the deeper the better.) I have not seen any indication that planning for subways in the US was ever bent towards the aim of being better shelters than they might be by default (which would involve "hardening" design decisions at the very least, storing supplies and careful ventilation choices at the most). But many discussions of urban Civil Defense in the USA did (and still do) talk about subways as one of many possible ad hoc shelters against blast or fallout. Again, as with the USSR, while US subways seem to be endlessly cavernous and accommodating, they could, even with lots of warning time, only accommodate a fraction of the total population of a dense urban area (as anyone who has been in these systems during rush hour knows quite intuitively!).

On the Moscow subway shelter estimates, see Edward Geist, "Was there a real 'mineshaft gap'?: Bomb shelters in the USSR, 1945-1962," Journal of Cold War Studies 14, no. 2 (Spring 2012), 3-28.