Blade Spam

A character attacks with a rapid series of slashes or stabs, either with a bladed weapon or with natural claws. Taking the speed part Up to Eleven and into outright Super Speed territory results in the following two subtypes:

Split-Second Blade Spam: Where the attacks are pseudo-simultaneous - the observer cannot tell that there are multiple slashes being executed.
Simultaneous Blade Spam: Superficially the same as Split-Second Blade Spam from the observer's POV, but explicitly being truly simultaneous.

Split-Second Blade Spam and Simultaneous Blade Spam are oftentimes the mark of Master Swordsman-type characters (or the equivalent for other kinds of weapon-users) in many works, and usually take the form of striking a given target from several different angles/directions - typically the eight cardinal and ordinal ones - within the span of a second.

Sub-Trope of Spam Attack. More Dakka is the Gun Counterpart, while Beam Spam is the laser/magic counterpart.. Compare Storm of Blades, for when several blades are rapidly thrown at the target.

Examples of Blade Spam include:

Standard Blade Spam

The Final Fantasy series give us Cloud Strife's Omnislash, Squall Leonhart's Renzokuken (specifically, the Lionheart Limit Break), and Tidus' Slice & Dice and the much stronger Blitz Ace.
A common Special Attack/Desperation Attack among sword/claw-wielding Fighting Game characters.
This is a staple in RPGs, usually part of Dual-Wielding and dubbed "Flurry". Examples to be found in Knights of the Old Republic and Dragon Age.
This is the default mode of attack for high-level Weapon Masters (a warrior Prestige Class kit) in Dungeons & Dragons. Able to fight equally well with both hands and using the attack/round tables of a Bare-Fisted Monk, a level 16 Weapons Master gets 6 strikes per round (about 10 seconds) with a single weapon, 12 when Dual-Wielding, and you do not want to get in her way when she's buffed with Haste.
The Akisazame (Tiger Teeth/Sword Rain) sword technique in the Tales (series), which makes the user stab at the enemy repeatedly, then finish with a final strike that varies from user to user.
Exalted: Extra Action charm + daiklave = lots of pain for someone.
Link can use the stabbing version throughout the Super Smash Bros. series, and gains the slashing version in the form of his Triforce Slash in Brawl.
Dante in Devil May Cry 3 has Million Stab (for Rebellion) and Million Slash (for Agni & Rudra) combos.

Split-Second Blade Spam

Claymore has two techniques that are this, Windcutter (which doubles as an Iaijutsu Practitioner technique) and the Quicksword.
Ginkaku Uneri from Katanagatari is an Iaijutsu Practitioner that can use his sword, Zantou Namakura, to strike five times a second.
In BlazBlue, Taokaka can do this with her claws.
Hiei's Curb Stomp Battle with Seiryuu in Yu Yu Hakusho. He cut Seiryuu in 16 places so quickly that Kuwabara couldn't see it at all, Kurama saw it as a single attack, and even Yusuke only saw eight slashes.
Sakuya Izayoi from Touhou is capable of Split-Second Blade Spam, as shown in her Wound Sign "Inscribe Red Soul" and Scarred Soul "Soul Sculpture" spellcards.
Fate of the Lyrical Nanoha franchise is capable of doing this with her Super Speed and Laser Blade. When she does it, the viewers will only see the aftermath of her slashes after she dashes past the target. Specific instances include her Speed Blitz against Numbers Cyborg Sette that left Sette's Battle Boomerangs in pieces, and her first attack on Huckebein's Deville that left him littered with slash wounds.
The Hiten Mitsurugi-ryuu secret move Kuzu-Ryuusen("Nine-headed Dragon") from Rurouni Kenshin, utilizing Hiten Mitsurugi-ryu's signature God-Speed to launch 8 slashes and one stab with such speed that the strikes appear to be simultaneous.
Rick Addison's blade spam is so fast that even The Hedonist Rance respects him.

Simultaneous Blade Spam

Two examples from the Fate/stay night franchise:
- Tsubame Gaeshi, signature technique of the 5th Grail War's Assassin; three slashes whose combined angles of attack explicitly block off any attempt at evasion. And Assassin gained it because he so relentlessly perserved in his countless attempts to devise a technique to kill a swallow while in flight, that reality basically gave in and granted him partial access to the "Second Magic", which is described as "multi-dimension refraction phenomenon".
- Nine Lives (Japanese gloss: "Shooting Hundred Heads"), a Noble Phantasm of the 5th Grail War's Berserker (a.k.a. Herakles); the actual NP itself, derived from Herakles' feat of defeating the many-headed, nigh-immortal Hydra with his bow and arrows, was never used by Berserker himself, but when Shirou in the Heaven's Feel route used Projection to adapt the NP for his own use, the resulting "Nine Lives Blade Works" was composed of eight simultaneous slashes from different directions in one instant, followed by a stab to the center, all done with Super Speed. And each strike was considered a lethal A-ranked blow to Black Berserker.
In more recent iterations of The Compilation of Final Fantasy VII as well as Kingdom Hearts and Dissidia Final Fantasy, Sephiroth seems to be able to strike his opponent several times with only a few tactically-placed slashes of his Masamune. In Kingdom Hearts II, he possesses a move called Flash, where he dashes past Sora; if you don't use the Reaction Command, Sora eats about 7-9 slashes. In Dissidia, his slashes inflict upwards of four blows at a time (his Octoslash, in particular, hits eight times in spite of Sephy only swinging five times).
The Toujin (Battle Blade) attack in Yaiba can do this, but with etheral blades which distract the opponent.

Mixed

In Soul Eater, Mifune can do this with his "infinite sword style" moves. Not only can he trap his opponent by pinning them down with some of his many swords but he can make each sword strike multiple times by making them vibrate.
Erza from Fairy Tail can do this with her exquip primarily with her heaven wheel armor.