JR East trains
No JR East trains allow unpacked bikes, but folded or packed bicycles (in bags) are usually fine, no additional fee required. Trains tend to be packed, so placing a bicycle in the very end of a frontmost or rearmost car is least troublesome. From the observation, JR Takasaki line is the least strict and an unpacked bicycle can be taken onto it without any trouble. The latter is rather a result of loose control and not JR East rule.
Narita Express trains belong to JR East but train configuration does not allow to put a bicycle with only one wheel detached inside. It is possible to place the bicycle into the aisle, but way for the drink selling cart may be obstructed. It is possible to place smaller bicycles into the baggage area.
JR Shikoku trains
JR Shikoku train lines do not charge an additional fee for a bike, but they are especially picky on what type of a bike bag you have. Make sure there are no uncovered bike parts sticking out of the bag - saddle included. Cover them with a plastic bag if no other options are available.
Hokkaido line allows bicycles to be taken as is, with additional charge, but there's somewhat loose control and it is not known how strict are bicycle bag regulations. No data for other JR Hokkaido lines.
Shinkansen (bullet trains)
As long as it is in a some sort of bicycle bag it is fine and no extra charge is taken. Put the bicycle bag behind the rear seat of a car. People carrying bicycles are common on bullet trains, so there's a possiblity you may need to check out several cars before finding a free place. The frontmost and rearmost cars are usually free. It is usually safe to leave the bicycle unattended. If you don't have a seat in the same car, it is okay to keep the bicycle near the door, as long as it is not blocking the passage between the cars. It is recommended to secure the bicycle/bag somehow, because Shinkansen tend to be surprisingly bumpy at times.
This depends on a particular ferry line, but most of the time it is possible to bring a bike with some additional fee. Ferries between Honshu and Shikoku island in Shimanami Kaido biking area take a bicycle for a basic fee. Most ferries in other areas allow you to secure your bike in the car area of the ferry. Bikes packed in bags can be taken for free.
City buses usually do not allow bikes onboard and in fact there's very little room to place them inside. Japanese buses do not have front-side bike carriers (as seen in the USA)
Tokyu lines, Keio lines, Odakyu lines (Tokyo area)
Same rules as JR East: no naked (uncovered) bikes, packed bikes are OK, no additional fee, any size.
Odakyu romance car
Packed bikes are fine with no additional fee. There's no space to place a bicycle but aisles are wide and putting them in the aisle is normally accepted by the staff.
Many rural private train lines allow taking bicycles onto the train with or without additional fee (depends on a particular line). Some lines limit the bicycle-free area for the suburbs only. Some lines have time limits (no rush-hour, for example), but on time-limited lines on the weekends and holidays bicycles can be taken on the trains with no limits. For the detailed information refer to the : Cycling Lines Wikipedia article (Japanese only).
In case of a large cycling event some lines allow taking uncovered bicycles onto a train for a limited period of time. Needs to be clarified right before the event.