Is there a online service which provides details of all the events happening in a city?
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The answer to this varies greatly from city to city. As a general rule though, no, there is no single service that can solve this (it’s actually an almost impossible problem to solve since events are by their nature decentralised).
It can be solved for specific categories of event. Songkick has extremely comprehensive gig listings, for example. Our startup, Lanyrd, aims to solve the sub-set of this problem that relates to conferences and professional events. We have pages for hundreds of cities, for example Conferences in London and Conferences in San Francisco.
Individual cities often have their own listings sites—often run by magazines or newspapers (TimeOut is a good bet for a lot of cities). There are also often niche-specific listings sites for e.g. tech events in a given area.