One my hobbies when traveling is recording TV video and data in cities I stay in. As such, I typically travel with a USB TV tuner, a mini-TV, and an assortment of small antennas. This includes on the train, though I seldom use any of this stuff on a moving train, because, as stated, trying to tune in TV signals in a moving vehicle is mostly futile and not really something I would recommend the average person attempt.
With that said, there are some situations where reception of a normal TV signal (i.e. not a signal specifically designed for mobile devices) in a moving vehicle can be surprisingly decent. By that, I mean a signal that only occasionally experiences breakups and is generally watchable for maybe 10-15 minutes or more. In general, such reception is possible in flat areas with limited trees/obstructions and a powerful TV tower close by. Stations that broadcast on UHF are also much more likely to come in under these conditions than ones on VHF (note that digital TV signals aren't necessarily broadcast on the channel number you tune your TV to - for instance, a station that you tune on channel 6, which is VHF, may actually broadcast on a UHF channel; channel 6 is just a "virtual channel" in this case).
Off the top of my head, I can remember being on the southbound City of New Orleans and being able to watch nearly all of the 10 PM news from WCIA as we cruised through the Champaign/Urbana area. I have also been on the Lake Shore Limited and Capitol Limited and managed to get roughly 10 minutes of watchable signal while passing through the Cleveland and Toledo areas. At least one of the Fargo TV stations should be viewable on the Empire Builder route between Fargo and Buxton, ND or so, too. I've never tried to pick up anything on the train there, but I have been successful in a car (not driving) going 75 mph on parallel I-29.
For reference, the antenna I've used for all of this is the simple portable telescoping one that Hauppauge includes with their TV tuners and also sells individually. If you have an electrical outlet close by, which you do on a train, a slightly larger amplified antenna may be marginally better, but not really worthwhile, in my opinion, since this whole endeavor is very much a novelty rather than something you should expect to work reliably or flawlessly.