Understanding how the repo market works has been covered by multiple online websites, including TalkMarkets, so there are a few repo centered websites listed towards the end of this article that would be helpful to readers interested in the subject. Most quotations below except the the discussion concerning the New York Federal Reserve Bank, come from a very helpful British repo educational website, Treasurers.org.

Repo is used by central banks, pension funds, sovereign wealth funds, money markets, insurance companies, banks, hedge funds, at bond auctions, in underwriting bonds, in secondary markets, etc. Before getting into basic issues with repo, we can see that bond yields trend lower in good times and bad, and bond demand in repo markets plays a big part in that demand.

The pressing question regarding the repo markets centers on collateral demand and scarcity. How much real and potential demand is there for sovereign bonds, used as collateral for all sorts of financial markets.

Hoarding of bonds in the repo markets may produce scarcity. It is not yet at the high level of scarcity that we could see in a financial crisis, if bond values weaken and/or bond demand for use as collateral picks up.

Sovereign bonds make up 2/3rds of the US repo markets, and that is 5 trillion dollars. Many times these deals are over collateralized due to risk, and so there is a possibility that the value of bonds used as collateral exceeds the 2/3rds estimate. There is no actual measurement of dollar value of these securities, but it could be developed in the future. Special bonds may receive fierce bidding due to their scarcity:

Market-makers and other dealers will use the repo market to borrow bonds that are in strong demand in the cash market (and therefore sometimes scarce) in order to fulfill delivery commitments on sales of those bonds in the cash market. One of the most common reasons for a bond to go special is when it becomes the cheapest-to-deliver in the futures market for that bond. Some futures sellers will have difficulty buying what they need to deliver to the futures clearing house. As failure to deliver to a clearing house would have serious consequences, these parties will be forced to borrow the bond in the repo market and they may have to bid aggressively to secure it.

Scarcity results in repo fails, a frequently covered on TalkMarkets by Jeffrey P. Snider and Tyler Durden.

The reason the repo markets are not larger is likely because of netting that offsets a large financial institutions' positions in the massive derivatives markets. However, measuring the size of the market is discussed at the end of this article. Non dealer activity is increasing and cannot be accurately measured at this time.*

Government bonds are the collateral of choice, used for the biggest transactions, and private corporate bonds, MBSs, ABSs, bank loans, and covered bonds are a smaller part of the market. Government bonds are used for their safety and liquidity. This is why tantrums are so often generated, in order to get people to sell sovereign bonds that are scarce, so the tantrum folks can generate a better interest rate. Sovereign bonds are superior to even gold. The new gold is, of course, sovereign bonds:

Gold. This is a very specialised type of collateral but its use has been boosted by the interest in gold generated by the crisis. 

One wonders if the Republican efforts to shut down government is not, in some way, a form of tantrum, so that big companies can hoard more bonds at better prices for their clients. It is not lost on many people that Ted Cruz, big on government shutdowns, at least as a threat, is connected to Goldman Sachs through his wife. I am not accusing, just contemplating that curious situation.

It will be interesting to see now that they are in charge, if the boys in Congress would actually push through to default. That goes way beyond tantrum to financial crisis. I would think that would not be pursued with a Republican president and congress, although POTUS once said he could negotiate bond discount deals. That would be a chilling destruction of collateral.

The repo market has the capacity to grow. It is smaller than before the financial crisis in the USA but not in Europe. Both are close to 5 trillion dollars in valuation and total world valuation is estimated at 15 trillion dollars. Repo exists because secured transactions are chosen by investors as the Great Recession put a damper on unsecured transactions not backed by collateral.

Since the crisis, because of higher risk aversion and regulatory pressure, repo has been attracting all commercial banks as well as a greater number of nonbank financials as such sovereign wealth funds, pension funds, insurance companies, endowments and corporate treasuries. 

All these organizations are likely hoarding bonds, big time. After all, repo exists to minimize settlement failures between counterparties. It doesn't always work that way, but it is hoped that repo is better than unsecured transactions of the past. Repo is a secured repurchase or collateral based loan to a counterparty. Ever larger liquidity positions are taken due to regulatory pressure. This increases the demand for bonds:

Collateral management is becoming ever more important. Demand for collateral for use in payments and settlement systems, as well as in the exchange traded and OTC derivatives markets, is being compounded by regulatory pressure on market users to hold larger liquidity reserves and make greater use of (collateralised) central clearing counterparties (CCPs), at the same time as a loss of confidence in some sovereign debt is creating uncertainty over the future supply of high-quality collateral.

So, there are two types of repos, repurchase agreements and sell buybacks. Both of these transactions act more like secured deposits by the way they are structured.  Estimating the value of the markets does not include another type of structured financial transaction.

In fact, repo markets are different than securities lending markets, so there are even more bonds used than  just used in the repo markets:

The collateral in securities lending can either be other securities or cash (securities lending against cash collateral looks very much like a repo).

However, equities are more often used in the securities lending markets.

From the article at treasurers.org noted above, we can see that scarcity is a real issue in the bond markets. The fact that scarcity is a possibility forces counterparties to hoard bonds. While repo markets reduce risk, they are not risk free. Again from the article at treasurers.org:

Collateralisation should not make lenders indifferent to the identity of their counterparties. This is because collateral is not perfect. The value of even the best assets fluctuates and the liquidation of even the best collateral in response to an event of default, particularly an insolvency, can be delayed by unexpected operational and legal problems. Consequently, collateral should be treated only as insurance against the default of the repo seller, not as a substitute for his credit risk. This means that the primary exposure in a repo remains counterparty credit risk.

Credit risk can cause counterparties to doubt a counterparty and to even refuse to accept collateral, preferring to pull out of the deal. Remember, collateral does not always cover the loss by the lender to the counterparty.

The second question regarding the repo markets revolves around rehypothecation. Firms that mismanage rehypothecation as did Lehman Brothers, will find themselves in a world of trouble.  But that is for another time.

For Further Reading (see also discussion regarding the N.Y. Fed below):

Defining Repo and Types of Repo Such As Triparty

Defining Term Repo

Repo Education

Too Big to Fail in Repo

Reference Guide to Repo and Securities Lending

Central Clearing for Repo and Derivatives

N.Y. Fed's Take on Size of Markets in Repo and Securities Lending

*Even the N.Y. Fed does not know how many securities are used as collateral, only the value of securities loaned out by dealers. Also, dealers are not the only participants in the repo market. Measuring other activity is not formulated. So, it appears the Fed is as much in the dark as the rest of us:

At present, dealers appear to represent the largest participants in the repo market. However, nondealer repo activity has likely increased, and without appropriate data collections, will be challenging to monitor. Advances in technology and changes in the regulatory landscape have made it more efficient for cash investors and collateral providers to engage in repo trades directly, bypassing a dealer. The nature of these nondealer repos is a brokerage transaction between cash investors and collateral providers.