Foldable bikes on public transportation

Are foldable bikes allowed on public transportation such as limited express trains, local trains, and buses? If so, what are some of the recommended foldable bikes? Not looking for anything too fancy. Just something easy to transport and use in areas where I can't get to via public transportation.

Trains, usually yes. Buses, usually no. But it depends on the policy of each company.

I remember a while ago reading the policies for some train companies. And the companies all shared very similar policies. I'm not 100% on this. But to travel on commuter style services, the bike needed to be foldable and be inside a bike carry bag to be allowed on trains.

Just googled this, it is recent:- http://www.14degrees.org/traveling-with-a-tern-folding-bike-on-the-tra...

1)
Learn/Search about "輪行(りんこう/rin-kou)".
https://ja.wikipedia.org/w/index.php?title=%E8%BC%AA%E8%A1%8C&oldid=60...
https://www.google.co.jp/search?q=%E8%BC%AA%E8%A1%8C&hl=en&biw=768&bih...
In Japan "サイクルトレン/cycle train" is very rare. (just not zero)
https://ja.wikipedia.org/w/index.php?title=%E3%82%B5%E3%82%A4%E3%82%AF...
(Use google translate)

Now most ExpWay/Exp bus(one door type) service dosen't accept by luggage trunk space problem.
Regular local 2 doors bus no problem execept rush hour in the morning/evening of weekday as manner.
https://www.google.co.jp/search?q=%E8%BC%AA%E8%A1%8C%E3%80%80%E3%83%90...
Cycle carrier/rack bus also very rare like cycle train.
http://www.city.atsugi.kanagawa.jp/shisei/15001/seisaku/bus/d015474.ht...

2)
Light weight under 12kg and compact folding type enough.
http://kakaku.com/bicycle/folding-bicycle/ranking_6440/rating/
over 20 inch tire with 6 or more than speed(derailleur) gears of Shimano.
https://en.wikipedia.org/wiki/Derailleur_gears
Shimano is not bike brand one of important parts brand.
A lot of major brands in and out of Japan using their parts.

A bike/cycle : 自転車(じてんしゃ/jiten-sha)
Folding bike : 折り畳み自転車(おりたたみ じてんしゃ/oritatami jiten-sha)
Carrying bag : 輪行袋/-バッグ (りんこう ぶくろ rinkou bukuro/-バッグ baggu/as roman spelling = bag)

Purchasing at Amazon or Rakuten web possible
but asking at large bike/cycle shop in/near yr town best.
Under 30000Y is fancy heavy weight(over 12kg) one not recommend if using a lot.
Under 10kg very comfortable but expensive one is target of theif.

(note)
In big city urban area has not enough parking(駐輪場/ちゅうりんじょう chuurin-jou) space usual.
So escape illegal parking(違法駐輪/いほうちゅうりん ihou-chuurin) need a lot of parking fee.
(fr 50Y and more/one time of one day. Not 1hr and 24hrs)
https://www.google.co.jp/search?q=%E8%B7%AF%E7%B7%9A%E3%83%90%E3%82%B9...
https://www.google.co.jp/search?q=%E8%B7%AF%E7%B7%9A%E3%83%90%E3%82%B9...
This is one of city,town bike law/rule guide.
http://www.kyoto-u.ac.jp/en/education-campus/international/students2/l...
If impounded you have to pay around 2000-5000Y for back/return.
(This is not "fine" as low but meaning mostly same)