The simplest chant is just the name of the team shouted over and over again, often with clapping in the gap; e.g. "Oxford" (clap clap clap); "Oxford" (clap clap clap). Chants being nothing if not competitive, opposing supporters may respond by shouting an insulting word in the gap.
The next simplest chant, used when your team is ahead, is just the score repeated, e.g. "two nil; two nil", sung to a tune approximating 'Amazing Grace'.
Chants can also support particular players. A common one is "One David Beckham! There's only one David Beckham" (or whoever). When an England international squad included two players both called Gary Stevens the chant became "Two Gary Stevens! There's only two Gary Stevens". Both are sung to the tune Guantanamera, as are "Sing when you're winning, you only sing when you're winning" and (when an easy shot or a penalty is missed) "Score in a brothel, you couldn't score in a brothel".
The song "Go West", by the Village People, provides the melody for the common chant "You're shit, and you know you are" and many, many others, including more specialised chants such as "One nil, to the Arsenal", "Go West Bromwich Albion" and "Posh Spice takes it up the arse", made famous when Victoria Beckham mentioned it in her autobiography as an example of the less-than-warm welcome shown to her by fans of Manchester United.
A further common format for chants is to sing a 7-syllable phrase to the tune generally associated with "You're not singing any more". Further examples include "Who's the bastard in the black?" (meaning the referee), "Can we play you every week?", "Are you X in disguise?" (where X is a weak or a rival team).
Some football teams also have songs which are traditionally sung by their supporters. Probably the most famous of these are Liverpool's (or Rogers and Hammerstein's) "You'll never walk alone" and West Ham's (or Jaan Kenbrovin and John Kellette's) "I'm Forever Blowing Bubbles".
Another chant is "Who ate all the pies"
Chants are less extensive in rugby union. England supporters sing "Swing Low, Sweet Chariot" for reasons which are not clear. (One widely held theory is that the practice originated at the match between England and Ireland at Twickenham in 1988, during which the former's Nigerian-born winger Chris Oti scored three tries.) The Welsh sing "Cwm Rhondda", which is the tune of the hymn "Guide me O Thou Great Redeemer". The New Zealand team (the All Blacks) are known for engaging in a ritual Maori haka before international matches. The Fiji team performs the cibi; the Samoa team the siva tau; and the Tonga team the sipa tau.
Chants are also used in other sports, including:
Rugby Union
List of football teams whose chants are described in Wikipedia