There is a significant amount of money in the stock market in US companies. For example, look at the largest US company by market cap - Apple. It is currently worth over 800 billion US dollars. I've heard there is about 1 trillion total US paper dollars. So Apple would cost you about 80% of all US paper money. I've also heard people say it is just 'funny money' and the 800 billion is not 'real money'. I would think there are brokerage accounts out there that account for the 800 billion that is Apple - that it is not just a made up number. So my question is, is there more money in traditional bank accounts like Bank of America checking and savings accounts or in brokerage accounts where the money in the account is accounted for by shares of companies and possibly other equities like ETF shares? What are the actual numbers? Is it even close? I see many of the big tech companies getting to around half a trillion dollars and it makes me wonder where all this money is when you also see articles that say something like 40% of Americans don't have $1000 in savings available for an emergency.
At the time of writing this answer, I would note that the question is somewhat confusing. The issue is that the question uses the common definition of the word "money" which corresponds to the term "wealth" in economics, while "money" has a narrow technical meaning (referring to the "money supply aggregates").
This section is the part that is confusing.
For example, look at the largest US company by market cap - Apple. It is currently worth over 800 billion US dollars. I've heard there is about 1 trillion total US paper dollars. So Apple would cost you about 80% of all US paper money. I've also heard people say it is just 'funny money' and the 800 billion is not 'real money'.
(Please note that I have not validated these numbers; the exact values are not important.)
Paper dollars (technically, "currency in circulation") is one form of financial instrument, and is a component of the various money aggregates. It is easy to work with, because it has a fixed value. The shares of a company (equity) are another financial asset, but they are not a component of the money supply. (That is, they are not "money" from the perspective of economics.) The value of equities change continuously during the trading day. If there is good news, the price of a stock could double, doubling the wealth of all of the holders. This means that a shareholder's brokerage account value could double, without any "money" changing hands.
Is this "real" money? It is certainly wealth. But it would be effectively impossible for all Apple shareholders to sell at the same time at the current share price, and so the market value cannot be converted into "money" at the current market capitalisation. This is not a problem for U.S. paper money - it can always be redeemed at par (for a bank account balance). Therefore, there is a logic to not treating the market value of a company as the same thing as the equivalent amount of money in the bank.
As for the comparison of sizes, any numbers I give will be outdated when later people read this answer. I will instead send you to a good source of data - the U.S. Federal Reserve's "Flow of Funds Report" (the Z.1 Report). Link to Z.1 Report
This report offers a quarterly summary of financial statistics for the United States. It contains data on flows - the value of transactions in instruments for the quarter, as well as balance sheet data ("stock variables" in economics jargon).
I would recommend just looking at the balance sheet tables for now. In particular, you can look at the table for "Households and Nonprofits" (table L.101). It has a breakdown of household (and nonprofit) balance sheets. (Link to html table for households.)
As of the end of 2016, households held \$15874.2 billion (\$15.87 trillion) in corporate equities; that does not include indirect holdings. Meanwhile, bank deposits and currency was only \$1237.2 billion, (or \$1.24 trillion). In other words, equity holdings swamp narrow "money" holdings (currency and bank deposits) for households. (Total bank deposits are much larger. I just wanted to compare direct holdings; the reader will have to look at the tables in their entirety to see total asset and portfolio sizes.)
The other balance sheet tables will also be useful to find the relative sizes of financial asset holdings by various sectors. This may help answer your concerns.
If all types of brokerage accounts are included, meaning stock, bond, mutual fund and call money, as well as IRA and defined contribution brokerage accounts, then the value of the brokerage accounts is somewhat greater than bank deposits.
The most recent analysis of brokerage accounts was publish at the end of 2015, Brokerage Accounts in the United States.
A total value of \$15,825 billion was determined for 2013 (the last year analyzed), though only \$4,274 billion was due to stocks outside retirement accounts.
The amount in bank accounts is much more well-known than brokerage accounts.
The FDIC publishes the amount of deposits in all FDIC insured institutions for 30 June of each year:
Keep in mind that a brokerage account is only one of 3 ways to hold stock, the others being a physical certificate and direct registration.
The "common man" conception of money is not helpful to think about these issues. Money is not ONLY currency and notes anymore. These are just a TOKEN used to distribute goods and provide exchanges. They are not the real stuff. For example, when you make a bank account transfer, you are not moving TOKENS. Why would you? Banks are given these tokens by the central bank in order to keep a smooth movement of payments.
So Apple does not have 80% of all tokens of the US economy in a vault somewhere. Why would they? It's dangerous! They have such ownership of wealth, but in the form of a current account and other assets.
So you need to think differently between a claim on wealth, WHICH CAN BE TURNED INTO TOKENS, and the TOKENS THEMSELVES.