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Examples (use Google or other search engine for details regarding the exchanges I'm about to list):

  1. Mt. Gox;
  2. BTC-e;
  3. Poloniex just after BTC-e went down which was also around the BCH fork;
  4. Plus many others which were either or both unreachable and/or folded at various times which didn't affect me. Yeah, I'm a slow learner. :)

Alternately: possession is 9/10 of the law. Why let a bank have possession of your resources? Same goes for exchanges.

Look into BitShares, which I also need to do; it's a decentralized exchange, so it's a wallet that you can trade in. Sounds great.

Also, use something like TradingView or Coinigy to do your trades -- it will use the API of the exchange that you have coins on. That way, if there's a rush into a coin (or out of...) and their HTTP server goes berserk, you'll still be able to buy or sell, since you're not using their HTTP server.

Good luck!