Alcohol and drugs
Cyclists who cycle on a road when drunk or high on drugs can be fined up to £1000. The test is whether or not you have proper control of your bike - there is no alcohol or blood limit per se.
Don’t assume that your driving licence will not be endorsed (or even revoked) just because you were found to be drunk on your bike and not in your car. The Power of Criminal Courts (Sentencing) Act 2000 enables courts to disqualify anyone from driving, without imposing penalty points, for any offence, including a cycling offence.
Careless or dangerous cycling
S.29 of The Road Traffic Act 1988 imposes a law against careless cycling (cycling without due care and attention or reasonable consideration for other road users) and this carries a fine (max £1000). S.28 of The Road Traffic Act 1988 imposes a max fine of £2,500 for dangerous cycling.
The test for these offences is the same as for driving i.e. did you cycle at the standard of a competent and careful cyclist?
More serious offences involving injury or death
A cyclist who causes injury or death by "cycling furiously" carries a penalty of a two year maximum imprisonment. This is covered by the Offences Against the Persons Act 1861.
Red lights and stop lines
If your bike crosses the stop line when the traffic lights are red is an offence under the Traffic Signs Regulations and General Directions, normally carrying a fixed penalty notice fine. Where there is an advanced stop line cyclists can stop ahead of the traffic but must still be behind the advanced stop line or they will commit an offence.
This is covered under very ancient Highway Act 1835 and is another example of how the law needs some updating. For example the Act does not refer to pavements, but to footpaths and says:
If any person shall wilfully ride upon any footpath or causeway by the side of any road made or set apart for the use or accommodation of foot passengers; or shall wilfully lead or drive any horse, ass, sheep, mule, swine, or cattle or carriage of any description […] every person so offending in any of the cases aforesaid shall for each and every such offence forfeit and pay any sum not exceeding …
Essentially, cycling on a pavement would be cycling "by the side of any road" and is an offence. In practice, the police will often exercise discretion and not rigorously enforce the law.