Dirty Dick's
One of the better known pubs although definitely no tourist trap near Liverpool St, supposedly named after a Georgian dandy who let himself go on the death of his fiancée.
Eat & Drink
A small and fairly ordinary Chinese restaurant by day, this turns into a heaving karaoke bar in the evenings. One of the most reliable places near Liverpool St to get a drink after midnight!
Lamb Tavern
One of several pubs in Leadenhall Market where you can listen to insurance brokers from nearby Lloyds talk business.
White Hart
An unpretentious City pub, slightly cheaper than most. Unusually for the area, has a few tables outside where you can watch the world go by in summer or cower under a heat lamp while smoking in winter.
Ye Olde Cheshire Cheese
An old City pub establishment, rebuilt shortly after the Great Fire of 1666. All the monarchs who have reigned in England during the pub's time are written by the main door.
If you just want to see the sights, come to the City whenever it suits you although check the opening hours for anything you particularly want to see. If you are spending more than a few days in London, visiting the area at night, especially around 10PM-11PM, can provide a decidedly un-touristy atmosphere. You will be seeing part of London life that few people who do not live or work in the City experience, and if you have the confidence to introduce yourself you may even get into conversation with local workers out for a late drink, the area is enough off the tourist route that you will be something of a novelty. Thursday and Friday are naturally busier but at the same time a bit less friendly, earlier in the week it may be dead but you have more chance of meeting locals just out for a drink.