The retiarii, or net fighters are a type of gladiator most are familiar with. They had very light armor (usually just an arm guard on one arm) and used the tools of a fisherman to fight: a weighted net, a dagger, and a trident. They are a bit of an anomaly in the gladiators world though, being based on an occupation (fisherman) rather than a type of military fighter. The fact the Romans considered this get up comedic speaks to the idea that in that era use of a trident (and net) were pretty unheard of in the military sense.
We can pretty safely rule-out their use in Northern Europe during the 8th-11th century as a weapon; none of the sagas that I'm familiar with mention a three-pronged weapon being used (and they cover pretty much everything, from clubs to horse-whips to swords), nor is there - to my knowledge - a word for 'trident' in Old Norse.
The Ranseur (Image no 7) (another page) is the only pole weapon I recall that resembles a trident. The hilt is crescent shaped and was probably used to parry and disarm opponents.
The book is An Illustrated History of Arms and Armour: From the Earliest Period to the Present Time, by Auguste Demmin.
The trident is a tool first and foremost, and its use in the gladiator's arenas has already been discussed and I have nothing to add to it that hasn't already been said. A typical trident consists of the trademark three tines, often barbed, and used primarily for moving fish in bulk similar to how a pitchfork is used to move hay.
There is however historical evidence of the use of "Military Forks" in battle, which bear a distinct resemblance to their three-pronged counterparts. The chief difference between the trident and the military fork being that the military fork typically only had two tines, and they weren't barbed (more tines and more barbs make piercing more difficult due to the distribution of force over more surface area).
The military fork saw use during the middle ages in Europe and was much simpler to use for lowly foot soldiers than a sword, which could take years to become proficient in. The two-tine arrangement made trapping an opponents weapon possible, giving it added utility when compared to a spear, though it seems to have been more commonly used in instances where a spear or other pole-arm was unavailable.
EDIT: Historical evidence of the military fork being used exists, in that we have found plenty of surviving intact artifacts, however what is less concrete is its actual usefulness as a weapon. There is conjecture on it being used for secondary functions as well (such as a tool for moving small loads of equipment, pushing ladders off a wall during a siege), but nothing concrete - but that's besides the point of the question in the OP.
Also, (I hope OP doesn't mind me piggy-backing off their question) are the tridents that were used historically flat, like we see in The Little Mermaid (so more like a devil's pitchfork) or cylindrical, such as the one used by one of the main characters in the Catching Fire movie / book?
According to Cruelty and Civilization: The Roman Games, tridents were used by a type of gladiator called a "Retiarius" in the arena during Ancient Rome. Retiarius i.e. "Net Fighters" would catch their opponents in nets and then use the trident while they were incapacitated.
I'm not aware of any specific uses in battle. A spear would be a much more efficient weapon to use - both in ease of production, ease of use, and the simple fact that a trident is not a good thrusting weapon due to its three-pronged design.
EDIT; Oops, I probably should have read the whole thread. Looks like my answer's been done.
Its only appearance for combat outside of myth would be in gladiatorial fights. In combat it was considered more interesting that gladiators have ineffective weapons to defeat their opponents, as it would be more entertaining and not to mention gladiators weren't killers and more resembled modern wrestlers. Tridents and nets are examples of such weapons and so one can assume they were ineffective in actual combat and war because of this place in the arena.
I'd also am curious about the connection between this discussion and the symbolism of the trident in Indian religions. I know the naga people carry a symbolic trident, and Naga is associated again with water beings. This imagery later went to Tibet and became a different kind of symbol in Tibetan Buddhism.
Also, as a follow up question: Are there any statistics as to how well trident wielding gladiators did against other armed foes?
In Japan, a variation of the conventional yari - spear - was the jūmonji yari, which featured a pair of slightly upwards-curved blades complementing the main point.
An European equivalent would be the partisan, which featured a very similar design. However, both the partisan and jūmonji yari aren't really akin to proper tridents - the two side prongs aren't nearly the size of the main spearpoint, nor to they curve upwards dramatically enough.
In the 'Romance Of The Three Kingdoms', several references are made to characters wielding 'Trident Halberds.' I'm afraid I don't have time right now to do extensive research into the Three Kingdoms era, but this is the head of a halberd from the Warring states era, about 450 years before the Three Kingdoms period. However, bear in mind that the Three Kingdoms was written in the fourteenth century, about events one thousand years before, and so any descriptions in it should be taken with skepticism, because the book frequently embellishes the facts, and some of the characters wield fantastical sounding weapons.
There is also a weapon called a tiger fork but it seems to have first been used by nineteenth century martial artists rather than soldiers in war.
The short answer is "no", but it's not really due to the history so much as the physics of melee combat.
I would define a trident to be a short weapon (under 6 feet), with multiple prongs designed for stabbing, with optional barbs on each spike. Longer weapons are not tridents, as they are considered polearms such as ranseurs, brandistocks, and others. Single pronged weapons are also not tridents, they consist of spears, pikes, and lances.
Single pronged weapons have much better piercing capability than tridents, as they have a single point to concentrate the thrusting power of a man's arm. This is a major advantage against armored opponents.
Against unarmored foes, this is a disadvantage, as it increases the chance of losing your weapon (as it gets caught in the body of your foe). If you are fighting a single combat (instead of warfare), that's not relevant, but spears frequently had a cross-guard to prevent this (see https://www.google.com/search?q=boar+spear&tbm=isch).
Tridents make it easier to hit your target (as you have extra prongs in case they dodge), but you're going to penetrate shallower (or not at all if they are armored).
Barbs are designed to entangle your target. This is mostly useful to keep prey from escaping when you are hunting. In warfare, there's a large risk that you will lose your weapon, leaving you unarmed against his buddy. For footmen facing armored cavalry, one tactic is to ensnare the rider with the hopes of knocking him to the ground. In that one scenario, you'll see barbs on weapons.
Short weapons are primarily designed to be used against men on the ground. If you are fighting a man on a horse, you want a polearm (if you are on foot) or a lance (if you are also mounted).
Multi-pronged weapons are mainly useful in warfare when they are polearms designed to unhorse riders by hooking or snagging them. If you are facing footmen, you're probably better off with a spear instead of a trident. If you want to kill the horsemen directly (instead of unhorsing them), you probably want a pike or a halberd instead of a ranseur or a brandistock.
