In the Czech republic, trains are one of the best ways to transport bikes quickly over long distances. But not every coach is suitable for every bike - recumbent riders with their long, wide and otherwise atypical machines know this very well. I recommend to buy tickets for Czech Railways (ČD) as early as possible via their e-shop. First because you are less likely to find all of the available places sold out. Second because it's cheaper. And mainly, you get to know what to expect: after clicking a detail view of a train, you can see transport mode and car types. There are also other railway companies than ČD, but I haven't tried them yet with a bike.
Speaking of tickets: ČD have a peculiarly complex system of discounts. Return trips, groups, early buys, action prices on some routes, students, seniors etc., plus you can buy a regular customer's card called IN 25, IN 50 or IN 100 (the number states your discount in percents, but they are pretty expensive and only pay off if you travel really frequently). Standard tickets are second class. You can also buy first class for some trains to get a bigger seat and better service, but it makes no difference to bike transportation options.
There are three modes of bike transport:
Nice and fast connection of Praha and Brno (and Břeclav and Austria). It tends to be very crowded on Friday afternoon and also full of bikers in summer, so be sure to reserve your place at least three weeks in advance. Ceiling height about 212 cm above floor. SWB recumbents don't fit the hooks, but you can park horizontally along the opposite wall equipped with two anchoring straps. Door width is above average, but there is a sharp turn around an electric cabinet right behind them, so prepare for some tricky maneuvering. Note: don't mix up with Regiojet, that's another company.
Doors are not so wide, but they lead directly to the cargo compartment without any turns, so you can wiggle throught them easily. Ceiling height 213 cm, emergency horizontal parking should be possible along the opposite wall.
Pretty much the same. Ceiling height 219 cm - a 'bent might fit vertically, depending on the front boom length.
Ceiling height 210 cm. A recumbent only fits horizontally; if not too long and doesn't block the door, it's OK, you just have to reserve a place next to the wall.
Tiny corner next to toilet, ceiling height 212 cm. 'Bents only fit transversally and then everything depends on how kind the conductor is: oversize luggage blocking passage between cars is not allowed, so you can be banned from the train.
I don't know if this car is intended for bikes at all. There's a bike symbol next to the door on the outside, but the small cargo compartment behind it doesn't contain any hooks and a horizontally placed bicycle would block all passages. So: folders only.
No compartments, no hooks, just a big multipurpose space the size of quarter of the car, present in several instances on every train. Bikes lean against folded seats and strap to place by safety belts. If the train is not overcrowded, even a tandem fits inside without any trouble.
Rather narrow corridor next to toilets is marked with bike and stroller symbols. One bike fits comfortably (although there is nothing to secure it in place), second one leans against the first, and I wouldn't guarantee third one fits.
Bike compartment with four hooks, usually occuppied by sitting passengers, is at each end of the three-car unit. Wide doors and easily accessible floor at the platform level, but ceiling height of only 200 cm. Most cyclists stay right behind the door and conductors don't mind.
Wide doors open directly to a low-floor compartment with four bike hooks. Horizontal parking is also possible, if there are not too many people. Ceiling is quite high, but I haven't measured it yet. Note: sometimes you get to your destination faster if you ride the bike instead of hauling it by this slow train.
Small motor train, still in service on some regional lines. A chance to board with anything larger than a folder only exists if there is a cargo compartment, which usually isn't. Better to ride the bike yourself.
Take an old 810, hang three bicycle hooks in the little compartment next to toilet, replace seats, give it a pseudo-English name and this is what you get. I don't know how many and how big bikes actually fit on those hooks and how they get inside through the narrow doors with steps and railings.
And finally some other company than Czech railways :-). Forget buying tickets at stations, conductor sells them on the train and even the most basic one is cheaper than discounted return-trip IN25 of ČD. The only disadvantage is that Arriva is not a part of PID (Prague integrated transport), so universal tram/metro/bus/ferry/train tickets don't work. Compartment with three bike hooks is at each end of the two-car unit. I wasn't able to measure ceiling height, but if you don't fit vertically, it's no problem to lean the bike along the wall, the train is usually not crowded at all.