Sword compilation

Koga
Probably the best in terms of mob clearing and certainly not a slouch in boss fights.
Her basic combo is 3 slashes which noticeably have very wide sweeps as well as good damage. Very useful for catching enemies.
Her signature skill is backdash coupled with a small AoE hitting where she was. While not incredibly impressive, it's a very useful escape tool which combos well with a gapcloser skill.
Her Ougi is an orb AoE centered around herself which multihits everything in it. Good for building combo, but honestly not very strong.

Tsuda
Probably the strongest of the original URs, while not as special as she used to be, she's certainly still one of the most useful swords in the game.
Her basic combo is 3 standard slashes. Their sweep isn't particularly wide, but Koga was the exception rather than the rule.
Her signature skill is a straight dash that does high damage and usually puts on her on the other side of whatever she's using it on. It can be quickly chained back and forth offering extreme burst damage.
Her Ougi is a targeted stunlock AoE that hits a specific enemy and those close to it repeatedly while Tsuda is still free to attack. It offers very heavy burst combo opportunities combined with her Signature.

Kou
A bit more restrained compared to the others, she has slightly lower damage comparatively, but a bit more utility.
Her basic combo is 3 standard slashes.
Her signature skill gives her a damage on touch effect letting her flinch enemies she moves into while also providing extra DPS on her combos which helps mitigate her lower damage.
Her Ougi freezes all enemies on screen while doing some damage. Some enemies like bosses though will just immediately shrug the freeze effect off.

Yoshimoto
Probably the least notable, but not actually bad or anything. Also I am over using the word probably I now realize.
Basic 3 slash combo.
Her signature skill is a flurry of wind slashes which knocks up the enemy into a down state. Can get an enemy off your back a bit, but not much utility or damage overall.
Her Ougi is an Excalibur style slash that has immense if narrow range. It's basically a straight line. It also doesn't really do too much damage considering the flashiness.

Yasutsuna
Insane damage for trivial effort. She has super armor and does not get flinched out of attacks. On top of that everything she does sets enemies on fire.
Her basic combo is 3 slow, but heavy hitting slashes. However she can instead hold the attack button to charge and move while doing so. One degree of charging results in a slash then an immediate kick. Two degrees/full charge results in 2 slashes then a massive damaging burst(18k damage) which even launches smaller mobs into other enemies. It also lights enemies on fire.
Her signature skill is probably the least notable part about her. It only gets 1 extra bullet from her skill page and is simply a knock down that lights the enemy on fire.
Her Ougi is her nuking the entire screen for 10k+ damage while burning everything. The last hit is an auto-crit that does 10k damage.

Momiji
The first hard hitter who actually sports decent utility.
Her basic combo is 3 slow strikes, but on the last attack, you can hold the attack button to grab an enemy. This locks her in place, but you can change direction. One degree of charge results in her throwing the enemy down at the floor in front of her. Two degrees/full charge results in her throwing the enemy at the wall of the screen.
Her signature skill is a strike that paralyzes. While it doesn't work on everything some bosses are indeed vulnerable to it.
Her Ougi is her least notable attribute. A simple launch then knock down into an AoE.

Sayo
Unanimously agreed upon to be the hardest to use UR sword for very little benefit. She's fast and can do a few neat things, but is simply to unwieldy and does not enough damage to compensate.
Her combo is 3 fast strikes that have very high movement, but short range. They can chain extremely fast and move her around very quickly.
Something most people don't realize though is that her combo can be continued into another attack right at the end. At the end of her combo a circle appears with 3 arrows, left, up, and right.
Swiping left or right at that moment will cause her to long dash forward and back for 2 strikes very quickly returning her to the same position. Swiping up will cause her to flip kick whatever she was comboing.
Her signature skill has her dash forward then flip kick repeatedly back to her original position. It brings enemies it hits backwards towards where you launch it from. Due to that it whiffs on enemies close to you and doesn't combo into itself well.
Her Ougi gives her a short super armored/defense buffed state that makes her initial strike a dash which then enters into her normal combo. This more than often completely moves you past the enemy you were trying to hit.

Izumi
While not quite as hard to use as Sayo, she's still up there for hard and unrewarding to use.
Her basic combo is 3 very far moving strikes. As in she'll move an entire screen in one combo. One of her biggest pros is she doesn't suffer movement lock ever while attacking. Other swords if starting a combo while still will stay immobile the entire combo. Izumi's nature however causes her to lunge constantly.
Her signature skill is a short lunge which knocks enemies up. Her standard attack has more movement though.
Her Ougi gives her 2 fire clones which mimics her attacks Ninja Gaiden style. While this gives her more DPS and more area coverage, it also lowers her base damage mitigating the DPS quite a bit. You do about only 33% more damage.

Kunohe
The weakest floater probably, but still incredibly useful since she's a floater.
Her basic combo is 4 very quick strikes. She movement locks easily however when not attacking an enemy.
Her signature skill launches a floating heart that slowly seeks and damages enemies multiple times. You can launch them consecutively and they'll float together, but they ultimately don't do much at all even together.
Her Ougi is a short full screen attack that hits many spots with hearts randomly, key word being randomly. You can attack during this, but it's not very reliable and only really useful for mob flinching. Not clearing as it doesn't do a lot of damage.

Musashi Masamune
She has 3 basic attacks that do a good amount of damage with the right equips. The last hit on her combo does the most damage but has a long recovery time so if you surrounded you might take a hit. Her signature skill lets her counter any incoming attack including projectiles and bomb explosions (although it's really dumb to waste a counter on bomb when you could just dash away). Her ougi does a combo with different types of elements and can burn. She's a great sword to use against bosses since she can stay on the boss and continue to combo and use counter when an attack is coming.

Koryuu
Has the potential to match or even surpass the limiteds. Her basic combo is 3 quick slashes that deal moderate damage. Her signature skill sends a small beam forward that paralyze anything that gets hit and survives. It can hit multiple targets if they're bunched together. Her ougi lets her create a clone of her self on the opposite side of her like a mirror. It might be hard to control but the clone copies any skill you use leading to some crazy damage. She needs very heavy investment to reach high damage since her base attack is very low but if you get the right gauntlets and if you have 100% crit skill you could probably melt a boss in a few seconds.

Sujingiri
She's pretty much a shotgun. Her basic combo is shooting 3 projectiles 3 times for moderate damage. She can clear a room without ever having to put herself in danger so she's useful for easy S rank. Her signature skill shoots a dragon head like a beam in front of her for ok damage. Her ougi turns the screen into a concert where all the enemies start dancing and launches them in the air at the end. Against a boss she can hit pretty hard up close but that puts her in danger unless you have a skill that lets her get away quick.

Kitsunegasaki
Basic combo is standard 3 slashes; they're a bit slower than some of the other "balanced" swords but they have decent damage and a bit more range to compensate.
Signature skill is a tail swipe that deals fire damage (lights weak enemies on fire as well). Important note: the skill attacks BEHIND you, but you can quickly turn in the middle of a combo and use the skill if you want to attack something in front of you. The swipe also has a wide arc so it's good to give yourself some breathing room as well.
Ougi is a bunch of fireballs that home in on one target and deal fire damage (these don't light the target on fire, though). Notes: the target must be in front of you. If there's only one enemy on screen and you're facing away from it and fire off the ougi, you'll whiff. The final burst once all the fireballs connect will knock the enemy into the air. Although the ougi is single-target, it does inflict AoE burst damage when it connects with the targeted enemy.
Overall, a good sword with good utility although probably not top-tier. Will definitely be highly useful if elemental weaknesses are utilized more by future enemies/bosses.

Kamewaritou
Basic combo is three SLOW slashes. Solid damage and extreme range, however (long sword + damage trails around the slashes). Notably, the third slash can be charged and held as long as you want. It reaches stage 1 in about 1 second and stage 2 in about 3 seconds. Going from no charge to stage 1 increases damage of the slash by about 325% (80 damage to 260 damage with no attack equips). Going from stage 1 to stage 2 does not increase damage any more, but it does add massive knockback to the slash, and any enemies hit while the target is flying back will also take damage (looks to be about 50%). Any further charging doesn't increase damage or knockback, although like I said you can hold it as long as you want without penalty and you can change direction while stationary. The lunge on the final slash travels half the screen on the practice stage. You CAN be staggered while you charge.
Signature skill is a self damage buff of about 150% that lasts for about 15 seconds. There's about half a second when it wears off where Kame will throw up and you can't move or attack. I'm not certain about this next point, but I think it also boosts critical hit rate. By how much, I can't say.
Her ougi is a multi-hit thing where she zooms around on screen hitting everything currently in the stage (I believe it hits things that might appear offscreen because the game slightly zooms out when you use her ougi). It does like 15 hits and then a final attack; the first set of hits does ~60% damage and the last attack does ~150%.
Overall, an extremely solid damage dealer for slower enemies (and maybe even faster ones if they're accompanied by slower ones) and in small arenas. Matches Momiji for potential usefulness against bosses, especially against those that summon a shit-ton of trash mobs.