What are the options for mutual fund supermarkets?
Which ones do people use and why?
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There are hundreds of entities which offer mutual funds - too many to adequately address here.
If you need to pick one, just go with Vanguard for the low low low fees. Yes, this is important. A typical expense ratio of 1% may not sound like much until you realize that the annualized real rate of return on the stock market - after inflation - is about 4%... so the fund eats a quarter of your earnings. (Vanguard's typical expense ratios are closer to 0.1-0.2%).
If your company offers a tax-deferred retirement account such as a 401(k), you'll probably find it advantageous to use whatever funds that plan offers just to get the tax advantage, and roll over the account to a cheaper provider when you change employers.
You can also buy mutual funds and exchange-traded funds (ETFs) through most brokerages. E*Trade has a nice mutual fund screener, with over 6700 mutual funds and 1180 ETFs. Charles Schwab has one you can browse without even having an account.