This did not use to be a problem when I first wrote Trading Online published by the Financial Times back in 1998! Back then, you just wanted to know the name of any broker.
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The biggest problem today is who can you trust. I get hundreds of emails a month from private investors asking me which brokers they can trust. The reason is technology has meant a proliferation of brokers and brokers realised that instead of matching buyers and sellers (like estate agents) there was more money to be made by being the house. They then realised there is more money to be made by:
widening the spread between the bid and offer ie buy and sell price
By moving the price so you could not exit at the ‘market’ price but only the brokers price
liquidating your positions by changing margin rules
Spiking prices to knock out your stop losses (usually late at night!)
And a growing number decided to just steal your money - they will call you up and say they will manage it, and then if they get lucky, they will not return it, and if they ‘lose’ then you have nothing anyway and the broker wins.
So the answer to your question is in my view, the same as finding a plumber! Go by the views of someone reputable and trusted in the industry ie word of mouth from someone who uses a specific broker.
I for instance was called by Bloomberg, who I trust as the most reputable brand alongside Financial Times. And since I used to present a trading online programme for three years on Bloomberg TV, the team knew me well. They introduced me to a broker, who previously I had not dealt with. And I said to Bloomberg, I would use the broker only if Bloomberg have done their due diligence on them. That is how I got to use and know one of the brokers I use. Yes, I am that careful. I need to know their CEO, go to their offices!