Please Note: Cardfight!! Vanguard is a game that is generally more balanced when it comes to different decks. Any cards that make it onto here are here for a very good reason. Even more so if the restriction/ban is seen internationally.
Barcgal holds the distinction of being the first card on the official Banned/Restricted list, and even then, it is only banned for use as a starting Vanguard. It's not so much Barcgal itself that is broken, but the fact that it can be used in a unstoppable chain with Flogal and Future Knight, Llew to set up a Turn 2 Blaster Blade with five cards in the Soul, a perfect set-up for Soul Saver Dragon and its Soulblast 5. This combo was so potent that it destroyed the Japanese competitive scene at the time, at least in the Senior Division.
Nubatama would have been worth a complete ban... had the clan been given more than four cards. The clan's entire premise is based around forcing the opponent to discard if they have more cards than you do. The only problem is that the hand is the primary source of defense. If this clan had ever been completed, it would have been more difficult to deal with than Barcgal-powered Soul Saver Dragon.
As of Booster Set 13: Catastrophic Outbreak, it is now possible to have a mono-clan Nubatama deck. However, they now have support that has been slightly toned down from their original strategy, going from outright discarding to binding with the potential for card loss, and currently only have one of each type of trigger.
They do have another critical trigger, however it is only given out as a promo card and thus difficult to obtain.
Eradicator, Dragonic Descendant got slammed onto the Restricted list a mere three months after its release. This may have had something to do with almost every professional Japanese player switching to near-identical Eradicator/Dragonic Descendent decks upon release. The card itself is absurdly effective, having 11000 Power (making it a good defensive card), and a self-standing ability that gives you two Twin Drives (all but guaranteeing at least one Trigger) and forcing your opponent to guard both attacks.
It's also worth noting that his self standing ability comes into play when he's guarded, AND giving him an extra critical to boot, making him deal two damage if the attack hits instead of one (and that's if the user doesn't get any other criticals during any of their drive checks). During the late game when this skill is most likely going to be going any hit from this guy will spell endgame for the opponent.
And even with being restricted to two per deck, Eradicator decks running Descendant are STILL topping in most tournaments in Japan.
It has finally been unrestricted.
Before Dragonic Descendant, there was the infamous Dragonic Overlord The End, which was the undisputed king of the Japanese meta for a year. Its self-standing Persona Blast allowed for two Twin Drives if the first attack hit, practically forcing the opponent to guard every single attack from The End. With a power of 13k (when crossridden) it was a powerful defensive Vanguard, and was backed up by a clan that could snipe any Rearguards who posed a threat. The End was finally stopped when it was put onto the Restricted list, and only came off in a (failed) attempt to wrest control away from Draconic Descendant.
Hot on The End's heels was Majesty Lord Blaster. Like The End, it could also force the opponent to defend every turn by having 2 Critical permanently simply by sacrificing two units, both of which could be easily searched from as early as Turn 3. Add onto the deck's capacity to retire important units with one of those pieces and you had a deck that could gain advantage ridiculously quickly. Like The End, it was soon restricted after a year, but unlike The End, it has still not yet been totally unlimited, with only two copies of it allowed in a Japanese deck.
And now, he's back to being unlimited.
Lizard Soldier, Conroe, widely regarded as the best starting Vanguard that the Kagero clan has ever received. The ability to add any Grade 0 or 1 to the player's hand at any time proved to be ridiculously versatile, especially after the metagame evolved to favour Perfect Defence cards (which are Grade 1s). The effect was good enough to make every other starting Vanguard, even those dedicated to supporting specific archetypes, sub-optimal.
Conroe eventually joined Barcgal as one of the only Grade 0s banned for use as a starting Vanguard. The tipping point was the rise of Dragonic Nouvelle Vague decks that would circumvent the deck's inherent Grade imbalance by using Conroe to search for Nouvelleroman Dragon, a card that would essentially turn all of your Grade 4s into Grade 3s.
Note that Nouvelle Vauge is the only legal Grade 4 card in existence, and only one other has been printed - and it can be argued that Nouvelle Vauge is highly superior.
It should also be noted that Mecha Trainer does the exact same thing as Conroe except he is used with the Spike Brothers clan. The only reason why Mecha Trainer has not seen a ban or a restriction at all is simply because the Spike Brothers clan simply does not have the wide and versatile Grade 0 or 1 pool that the Kagero clan has. In other words, Conroe is banned and Mecha Trainer is not simply because the Kagero clan is better.
Cat Butler has been restricted to two in Japan due to its interactions with Legion vanguards such as Mega Flare.
The Thing Saver "Abyss" deck. A variant of Royal Paladin decks centered around Seeker, Thing Saver Dragon that used the Shadow Paladin legion unit Phantom Blaster "Abyss" to build up soul, so that Thing Saver's skill could be used twice per turn. It consistenly topped tournaments, causing Bushiroad to put a term to the deck's existence by changing the rule that allowed Royal Paladin players to have up to 10 Shadow Paladin cards in their deck to a new one that allows them to have up to four "Blaster Dark" in their decks.