In all the parisshe wyf ne was ther noon That to the offringe bifore [the Wife of Bath] sholde goon. (General Prologue 451 – 452)
The Wife of Bath is in competition to be seen as the most important woman in her parish church. Not only does this provide us with an important piece of information about the Wife's character, it also tells us about what could serve as a status-enhancer for a medieval woman: in this case, being first to give the offering at mass.
Over al ther [the Miller] cam At wrastlyng he wolde have alwey the ram. (General Prologue 549 – 550)
Compare the prize the Miller wins in his competition – a ram – with what we learn the Knight wins for his prowess in battle – a great reputation. These different prizes might represent the different registers in which the Knight and Miller are portrayed – the Knight's is abstract and noble, the Miller's is earthy and animalistic.
If that he faught and hadde the hyer hond, by water he sente hem hoom to every lond. (General Prologue 401 – 402)
It's interesting that among the first competitions we hear about in The Canterbury Tales are the Shipman's physical fights that end in death by drowning. This colors our perception of the conflicts between the pilgrims that follow, some of which also threaten to become physical and violent.
At many a noble armee hadde [the Knight] be, At mortaille batailles hadde he been fiftene And foughten for oure feith at Tramyssene In lystes thries and ay slayn his foo. (General Prologue 60 – 64)
Chaucer is careful to emphasize that the Knight fights for the Christian faith. He does not engage in competition for no reason whatsoever, like some of the other characters we meet in the Prologue.
And which of yow that bereth him best of alle, That is to seyn, that telleth in this cas tales of best sentence and most solas, shal have a soper at oure aller cost. (General Prologue 798 – 801)
The Host is a lot more specific about what will win the competition than just, "the best tale." The winner has to tell the tale "of best sentence," that is, the best instruction, and "most solas," the most delightful. This gives us an idea of what the Host thinks makes for a good story, and it corresponds with ancient Greek philosopher Aristotle's definition of what the purpose of literature is – to instruct and delight.
The Reeve's Prologue
And, by your leve, I shal [the Miller] quyte anoon; Right in his cherles termes wol I speke. (Reeve's Prologue 63 – 65)
The way the Reeve's going to "quite," or get back at the Miller, is by speaking in the same terms he does. This sets the rules of the competition between them as a sort of "copycat" game, and gives us one way of looking at the Reeve's tale – that is, in comparison to the Miller's tale.
For leveful is with force force of-showve. (Reeve's Prologue 58)
This is kind of like saying, "you've got to fight fire with fire." And by saying this, and therefore excusing himself for speaking rudely, the Reeve makes himself appear better – more polite than – the Miller (i.e., the Reeve is only speaking rudely because he's got to fight fire with fire, whereas the Miller is just rude by nature).
Introduction to the Man of Law's Tale
[. . .] I recche noght a bene Though I come after hym with hawebake, I speke in prose, and lat him rymes make. (Man of Law's Introduction 94 – 96)
The Man of Law explicitly sets himself up in competition with Chaucer. In contrast to the Reeve and the Summoner, though, who want to fight fire with fire, the Man of Law proposes that the best way to overcome an opponent is to do something different than him and avoid the competition altogether. So, he announces he will speak in prose, rather than attempt to beat Chaucer at his own game.
The Friar's Prologue
[. . .] Whan it comth to my lot, By God, I shal hym quiten every grot. (Friar's Prologue 77 – 78)
The Summoner allows the Friar to say what he wishes, because he knows that, later, he will have a chance to speak and get back at the Friar. It's interesting how, here, anticipated future competition is actually a means of preventing conflict in the present.
The Franklin's Interruption
For he to vertue listeth nat entende; But for to pleye at dees, and to despende And lese al that he hath is his usage. (Squire's Tale / Franklin's Interruption 689 - 691)
The Franklin's son's gambling is the kind of competition that wastes money and serves no purpose. It signifies vice, in contrast to the Knight's fighting for the Christian faith, which signifies virtue.
Men shal nat maken ernest of game. (Miller's Prologue 78)
Here "game" can mean either something funny, or it could be referring to game in the sense of competition. By calling his repetition of the Miller's words "game," Chaucer compares his narration to the tale-telling competition in which the pilgrims engage.