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I am moderately claustrophobic. The idea of riding a jammed packed subway in Tokyo makes me nervous.

But I plan on riding the subway in NYC. Are there certain trains that are more busy and certain trains that are less busy.

Are peak times the standard rush hour times?

More importantly, is there any resource where I can see a comparison of the rider congestion at certain trains or stations?

The New York City Subway, while not at Tokyo or Hong Kong levels of crowding, is definitely crowded; with the population rebounding in recent years after decades of decline, ridership in the system is at its highest since 1948. The much-ballyhooed opening of the Second Avenue Line has relieved pressure on the Lexington Line, but of course it's adding more people to the system overall.

While statistics about the busiest stations are not hard to come by, this is not necessarily a reliable guide to determining which stations will be more crowded. After all, the busiest stations are also some of the largest. Times Square-42nd Street sprawls out over a city block with five different sets of platforms, and that might mean that any given part of it is less crowded than, say, 18th Street on the Broadway Line and its narrow split platforms.

Similarly, the architecture of a station may make it feel more crowded than another station of similar size, as there is some variation in the number of support columns, the width of the platform, the height of the ceiling, the existence of a mezzanine, the look of artwork on its walls, and so forth. An elevated or surface line has obvious advantages over an underground line as well, and so might tip the more crowded 7 Train over the less crowded F train in Queens.

For the trains themselves, you might have a look at the annual State of the Subways Report Card published by the Straphangers Campaign (straphanger being a nickname for a subway commuter), a project funded by NYPIRG, an activist outfit. The report provides assessments of twenty key lines on measures like reliability and cleanliess. The December 2016 report ranks the R train as least crowded and the 4 train as the worst.

Peak times are the same as in most major cities— from about 7am to 10am and from about 4pm to 7pm (with some variation by line). There is plenty of movement in all directions, though naturally there will be more flow out of predominantly residential areas (e.g. uptown, Queens) to midtown and downtown in the morning, and the reverse in the evening. Events like baseball games or festivals may also affect how crowded a particular line or station will be on a particular day.

Are there certain trains that are more busy and certain trains that
are less busy?

It depends a lot. All trains are busy but only in certain areas. Usually trains are less busy first 3-7 stops from the last/first stop (local). For express train first 2-5 stops can be ok.

Are peak times the standard rush hour times?

I recommend you to ride the subway around 10:30AM-2PM and 8:30PM-12AM. Much less people and completely different experience. I was on the 1 train going uptown from 168th St completely alone in a car. It was around 1PM.