New York - NYC

Not sure how the London letting scene is nowadays, but NYC is a very tough nut to crack. I've just moved here after two years in LA, and finding a place was a proper pain in the arse. The below advice contains a bunch of stuff I wish I'd known two months ago.

It's all about your credit history here, and if you're fresh off the boat you won't have any. However, zero credit is much better than 'bad credit'!
You will also need to prove your income - as a guideline, your gross annual income should be 40-50 times the amount of monthly rent you want to pay. Any less and the landlord will think you won't be able to keep up with the rent.

Depending on your situation (age, employment, etc) it might be advisable to start searching amongst NY friends if you have any, and be open to the idea of initially sharing an apartment with somebody you know (or a friend of a friend) whilst you build up a credit rating.

The Broker thing is typical Estate-Agent bullshit. You DO NOT need your own Broker, as the majority of properties advertised on the web are repped by a Broker of their own anyway.
If you show up with your own Broker in tow (as we did), the Broker repping the property will immediately realise that they're going to have to split their fee 50/50 with your Broker if your lease application is accepted - they call this a 'co-brokering' situation. This, along with a lack of credit history will put you WAY down the desirability list.
Broker fees are typically 10-15% of the annual rent of the apartment you're looking at. An effing big chunk of money. Can be avoided if you move into a spare room at a mate's place, or find a 'No Fee' listing on sites like the aforementioned Padmapper or StreetEasy.
Also worth asking around the coffee shops in neighbourhoods you want to live in about where to find no-fee apartments for rent.

Open-house viewings are probably the most unpleasant thing - 40 or so people trying to view the same apartment at the same time. Usually on a Saturday. I fucking hated those.

To give you a little perspective on what it was like for us just two months ago;

• Wife works for a major American company, on a good wage.
  
• No US credit history as we spent the two years previous living debt-free in LA, owned our (cheap) car outright, no credit cards, etc.
  
• Lived in same apartment for 2 years in LA, had glowing written recommendation from old landlady.
  
• Wife's company assigned a Broker to us. Unfortunately he wasn't even vaguely familiar with the areas we wanted to look at, and we ended up feeding him listings most nights of the week.
  
• In 11 days we viewed 29 apartments, and on just two occasions we met the owners. The rest of the time, it was just us and Brokers.
  
• Applied for 4 apartments and were rejected from 3. We never found out why we were rejected from any of them, and our Broker couldn't tell us either. The application that was accepted was from a viewing where we met the owner (and more importantly, they met us).
  
  

If you go down the road of finding your own apartment and paying a Broker Fee, be certain that you're getting into a place you really want.
A few days of NY apartment-hunting should be enough to convince you that you sure-as-shit don't want to be looking for a new apartment again in a year's time.

HTH