What planeswalker cards can be used as commanders in the Commander format?
Currently (in November 2018) there are the following sets of planeswalkers that you can use for your commanders:


The five double-faced transforming planeswalkers from Magic Origins. This is because they're a Legendary Creature on their front face, which is what the Commander rules are looking for.
Nicol Bolas, the Ravager, also a double faced card that starts as a Legendary Creature.
Three sets of planeswalkers which say “(This card) can be your commander”:


The five mono-color planeswalkers from Commander 2014
The four multi-color planeswalkers from Commander 2018
Rowan and Will Kenrith from Battlebond



These latter sets of commanders are still not legendary creatures, but golden rule 101.1 lets them make themselves exceptions. 


  101.1. Whenever a card's text directly contradicts these rules, the card takes precedence. The card overrides only the rule that applies to that specific situation. The only exception is that a player can concede the game at any time (see rule 104.3a).


This is like how the Commander 2016 Partner commanders also get to make themselves exceptions from the usual Commander rules.



As of Ixalan, all planeswalkers are going to be Legendary Planeswalkers, including retroactively. You still won't be able to use them for your commanders because they won't be Legendary Creatures. Sheldon of the Commander Rules Committee has said there aren't plans to allow all planeswalkers (posted 28 August 2017, in response to an inquiry about planeswalkers becoming legendary):


  No, PWs will not be allowable as commanders unless a card specifically says that it can--so no change.


(This is like how they're not allowing Legendary Artifacts or Legendary Enchantments — it's not about being legendary, it's about being a Legendary Creature.)

House ruling exceptions for fun

That said, Commander is intended to be a format played casually for fun with friends &mdash; the official EDH site's Philosophy section says as much. That same section explicitly encourages people to make exceptions or changes to the rules for local play if it's fun. Unless you're in some kind of official sanctioned event, adherence to the Commander rules is only necessary insofaras you and your friends find it fun. If you and your friends want to allow Planeswalkers as commanders, by all means, talk about it and create such a rule between yourselves, and then make a deck with a Planeswalker as a commander.

Where are the rules for planeswalker not being commanders?

There's two sources of Commander rules you can check to verify the absence of Planeswalkers in general being allowed:


The official Commander/EDH site's rules, which say the following:


  Players must choose a legendary creature as the "Commander" for their deck.


No mention of Planeswalkers.
The MTG Comprehensive Rules, probably easiest found on the MTG Wiki Commander format article, which also includes some community commentary and explanation. Those rules say:


  903.1. In the Commander variant, each deck is led by a legendary creature designated as that deck’s commander. The Commander variant was created and popularized by fans; an independent rules committee maintains additional resources at MTGCommander.net. The Commander variant uses all the normal rules for a Magic game, with the following additions.


(Notably they defer to the site I linked in point 1.)