What are the different designs of lightsabers in the Star Wars canon? For example, Darth Maul with his double-bladed saber, and Kylo Ren with his cross-guard saber.
Double-bladed lightsaber (see: Darth Maul, Pong Krell, Savage Oppress)
While I hesitate to call this a completely different style, Master Krell's lightsabers are unique in at least one way: they fold up
This is no longer quite as unique as I thought; in the Star Wars: Rebels episode "Shroud of Darkness", Kanan duels with a Force Vision of a Jedi Temple Guard, who uses a folding saber.
Curved-hilt lightsabers, like Dooku's:
Asajj Ventress' lightsaber(s)
Ventress wielded two curved-hilt lightsabers. In The Clone Wars animated film, she was able to combine them into a single, S-shaped double-bladed lightsaber:
However, Obi-Wan Kenobi possibly destroys the mechanism that allows this during his fight with Ventress in the film; he cuts her (connected) lightsaber in half, and we never see her use the technique in The Clone Wars show.
Lightsaber pike, used by the Jedi Temple Guards1:
We have a close encounter with a Temple Guard in "Shroud of Darkness", a Star Wars: Rebels episode, where Kanan has a duel with (a Force ghost of) one using a folding lightsaber. It's not apparent whether this is a standard Guard's saber or not, but there is the reasonable possibility that these sabers do fold.
Crossguard lightsaber, most famously used by Kylo Ren:
But also in use by ancient Jedi; Ezra Bridger finds a crossguard saber on the Sith world of Malachor, a relic of an ancient battle:
Shoto (a shorter hilt, used by smaller Jedi like Yoda and Even Piell)
Shotos are also used by the rare dual-wielding Jedi as an off-hand lightsaber; in the below image from The Clone Wars, you can clearly see that one saber is dramatically shorter than the other:
I regret that I do not know what episode this image is from; I welcome input from anyone who does.
Sabercane (Cane/lightsaber combo), used by Master Tera Sinube:
Lightsaber/blaster combo (invented by Ezra Bridger, apparently)
Not merely a double-bladed lightsaber with a funky grip, the saber style used by the Imperial Inquisitors spins around that central ring. The image above is of the Grand Inquisitor, from season 1 of Rebels, but we've seen the other inquisitors in season 2 using sabers of the same style
Debated whether or not to include this, but Kanan Jarrus' lightsaber exhibited a property we've never seen before:
It comes apart into two pieces. There a bunch of things we don't know about this lightsaber, including whether the pieces function without being connected, or if this is something all lightsabers can do but we've never seen before, but it's an otherwise unique property
Most of the above, plus:
Guard shoto (or Lightsaber tonfa)
One-bladed variant on the lightsaber pike:
Whatever you call this thing:
Thunderforge, in comments, suggests three-bladed lightsaber because there of the third blade in the middle of the handle. Wookieepedia calls it the "War Dragon rider lightsaber", because its only appearance in any material is a "design an alien" contest in Star Wars Galaxy Magazine. The War Dragon Rider actually won the contest, no doubt because it looks like something a Space-metalhead would have airbrushed on the side of his spacevan.
I believe Ventress' lightsabers were also slightly more interesting in Legends; in addition to the weirdness stated above, I believe she could tie them together with cord and whirl them around like nunchaku; I can't recall seeing this in canon, but I'm happy to be corrected.
The earliest true lightsabers are now typically called retrosabers; their distinguishing feature was an external powerpack, typically worn on the belt and connected to the hilt by a cord.
After the Jedi went wireless, protosaber look-alikes were called "retrosabers," which also used the external power source, and continued to be used by Jedi hispters the galaxy over (the above image is from an RPG sourcebook set during the Clone Wars)
The earliest iteration of the lightsaber was called the Forcesaber; it was used by the Force Hounds, Force-sensitive hunters used by the Rakatan Infinite Empire to find worlds with lots of Force-sensitive inhabitants (which the Rakatans would then invade and enslave). The main difference between the Forcesaber and a lightsaber is one of engineering: rather than requiring a battery, the Forcesaber is activated by channelling Dark Side energy through the hilt.
1 My first instinct was to classify these as double-bladed lightsabers, but they have their own entry in the Databank, so ho-hum
There are lots of lightsabers:
Double-sided: a lightsaber with blades on both sides. Some can be taken apart into dual lightsabers. Some models can fold up. Typically longer than a normal lightsaber. Can be used as a long-handled single bladed lightsaber. Some notable examples of use were by Asajj Ventress, Darth Maul, Pong Krell, and Jedi Temple Guards.
Dual: Two lightsabers, normally alike. Some can be put together to make a double sided lightsaber. Some notable examples of use were by Asajj Ventress, Ahsoka Tano, Starkiller, and Anakin Skywalker (just for a short time on the Trade Federation Control Ship).
Crossguard: A single-bladed lightsaber with two mini-blades protruding from each side of the handle, normally making it slightly longer than normal. Better for blocking enemies with more saber range. A notable example of use was by Kylo Ren, though a few ancient Jedi and many ancient Sith used them.
Shoto: A shorter version of the lightsaber, with the handle a few inches shorter and around a 2 1/2 - 2 foot long blade. Some notable examples of use were by Ahsoka Tano and Yoda.
Lightwhip: A type of single blade lightsaber except the blade was like a whip, so only extremely skilled Jedi or Sith could wield it without killing themselves. I don't know of anyone who uses them, though.
Spinning: A double sided lightsaber that can spin on a track built into the handle, so it basically spins automatically. It can be used as a single-bladed spinning lightsaber, though, too. Some notable examples of use were by The First through Thirteenth Sisters (Estimate) and the First through Thirteenth Brothers (Also estimate).
Dual-Phase: A lightsaber that uses two Kyber Crystals, therefor being twice as long. Some dual-phase lightsabers can be switched between 3 feet and 6 feet. I don't know of any users of this blade.
Curved hilt: A lightsaber with a curved hilt, enabling much more maneuverability. Sometimes used as dual and dual curved hilts can be combined to make a double-sided blade with an S-shaped handle.
Blaster-Saber: A lightsaber with a blaster attached, enabling you to have a compact weapon. A notable use of this type was by Ezra Bridger.
Underwater-Saber: Most lightsabers short out when they touch water, but underwater sabers don't. The only known one was used by Kit Fisto.
Lightsaber Pike: A type of lightsaber, typically with a very long hilt, long than a double sided hilt, and sometimes a longer or shorter blade. I believe some Jedi Temple Guards used them.
Darksaber: A lightsaber with a rectangular prism hilt instead of a cylindrical hilt. The blade is shaped like a sword's blade, and is a black flecked with white color.
Normal single bladed: Most Jedi and Sith use this. It's in balance with all the pros and cons.
There are others that I haven't covered here, but I believe I got most of them.