Japan Guide goes into which trains offer unreserved seating and which don't:
Most shinkansen trains offer both non-reserved seats (自由席, jiyūseki) and reserved seats (指定席, shiteiseki) in separate cars. Only the Hayabusa, Hayate and Komachi trains on the Tohoku Shinkansen and Hokkaido Shinkansen and the Kagayaki trains on the Hokuriku Shinkansen are fully reserved and do not carry non-reserved seating.
If you're not sure, you can check the train you'll be taking ahead of time on HyperDia.
If there is an "unreserved seat" option under "seat fee", then that train has unreserved seating. Otherwise, it doesn't. Note that you still have to pay a "seat fee" even for unreserved seating. This is because the "fare" is actually the same as a regular train, and then the "seat fee" is a combination of the "limited express fee" and seat reservation fee.
Note that if you're on holiday in Japan and travelling with the JR Pass, you can request a free seat reservation by visiting the ticket desk at the station.