I think the best way of analyzing your question would be to break the destruction of the Bitcoin into categories. There is the notion of a hard destruction meaning an attempt is made to physically compromise the Bitcoin network either by a 51% attack or international legislation. And there is the notion of a soft destruction, where attempts are made to De-legitimize the currency through media manipulation and also persistent DDOS attacks on the infrastructure supporting the Bitcoin.

In terms of Hard Destruction, I can identify the following scenarios:

1. The NSA or some other entity with both the budget and experience create a VLSI project to both develop and deploy an ASIC design that would result in a 51% attack

2. International regulation is developed that significantly inhibits one's ability to exchange Bitcoins for local currencies. Essentially forcing the Bitcoin underground like a drug cartel

3. A mathematician discovers how to break ECDSA (very unlikely)

4. Innovation results in the Bitcoin being replaced with another currency

In terms of Soft Destruction, I can identify the following scenarios:

1. The media alongside a covert multi-government effort conduct several propaganda campaigns to sway public opinion that the Bitcoin is either a massive scam or somehow bad

2. Cumbersome regulations are adopted to monitor and control Bitcoin exchange

3. Persistent DDOS attacks occur on the major exchanges like Mtgox and also the supporting infrastructure

4. One way to attack Bitcoin would be to have large amounts of money alternately pushed into Bitcoin and pulled out of Bitcoin, thereby massively increasing volatility. These market fluctuations could be aggravated by a covert government programme of destructive funding and public dis-information. This would make doing business in Bitcoin more difficult.

In all honesty, the Bitcoin is an experiment. We have no idea what will happen and who will oppose it.

Honestly, I think the best way for the government to shut down Bitcoin would be as a secret project. Simply construct a VLSI ASIC miner built using 40nm/1Gtransistor technology and build 100,000 of them. Effectively make them appear distributed around the world using VPNs, the bandwidth is low. Then execute frequent 51% attacks on the Bitcoin chain to "unconfirm" transactions to major sites. Confidence in the Bitcoin system would drop to nothing.

Another trick would be to switch the miners off and on unpredictably. If done right, this could lead to hours between confirmations at some times.

One can imagine the community responding to this in a coordinated way to develop a workaround. But if you assume subversion and interference in that process, such a response could be avoided.

Of course, they could also summarily execute anyone found using Bitcoins. It's effective too, but also pretty unlikely.

My predictions are they cannot close it down, but they can undermine confidence by: accepting anti-Bitcoin law, closing exchanges, closing Bitcoin businesses (according to law), infiltrating Bitcoin client developers, attacking the network (this is very extreme, low probability of happening).

By publicly attacking Bitcoin, they would feel the heat from people. Governments around the world are currently loosing support on every step they make and this would be no exception.

Edit: they CAN close it down by shutting down the internet (most extreme, government would most probably fall as a result)

It seems to me that this question can be answered by substituting Bitcoin for similar technology and re-posing the question. For example: "How easily could the U.S. Government shut down bittorrent" is quite similar and has already been answered. The U.S. Government has TRIED to shut down bittorrent and its associated sites a great many times. The Government knows with 100% certainty that bittorrent is not only being used for illegal activities but also knows that its use for illegal activities greatly outweighs its use for legal ones. There are many good comparisons worthy of note, but here's the big one: The U.S. Government hasn't gone after bittorrent itself.

The reason for this is twofold: First it is both difficult and unreasonable to outlaw something simply because it could be used in an illegal manner. Bittorrent in and of itself is a tool, like a screwdriver and as such cannot reasonably be declared illegal. As a competing currency, Bitcoin may be different enough to be vulnerable in this aspect. The second and far more important reason is that they simply can't. Distributed protocols like bittorrent or Bitcoin make it almost impossible to shut them down. For every governmental countermeasure there is a personal countermeasure and it's easier to find the actual criminals if you haven't driven them all to TOR or I2P.

The government cannot shut down the protocol. They are aware of this and instead pursue individual criminals. Attempting to shut down the protocol is futile and only makes it harder to catch individual criminals.

In many cases a government could effectively shutdown bitcoin in its country by making the following law:

No licensed or regulated financial institution can exchange currency for bitcoin.

Since taxes and fees for government services must be paid in currency, if I receive bitcoin for services, I would have to exchange the bitcoin for something else that could then be sold. A country would have to be extremely messed up for a significant number of people to want to do this.

And in countries where this did not work, they could forbid the usage of bitcoin; which would make using bitcoin on the black market difficult as you could only barter in none transparent black markets situations.

They could DDOS the network since the blockchain can only handle about 3 transactions / second if a party (Or group of parties) were to generate say 6 / second on over the course of weeks or months it would pretty much make it so no one would be able to have a durable transaction.

Easiest way is to kill power/internet that the network so desperately needs.

Nothing need be done technically. Just economics. Supply and demand. Government has infinite resources. With unlimited use of the money supply, make the system so volatile as to loose favor with the Bitcoin network. Corner the market, then crash it for instance.

Could be pretty easy. At least to cause people to pull out of bitcoin. If a government setup a massive covert trusted exchange platform, such as mt.gox, or bigger. They could have tons of coins being processed at any given time. Then the exchange shuts down, not unlike mt.gox. If the mt.gox disaster happened today with such a high value on the coins ($4000usd+), maybe even on a bigger scale, also assuming many people may have actually paid cash to acquire these bitcoins, it would sink it once again. If you manage to take your money out at the right time, thats great. But to store them away by allowing people to purchase goods from you using bitcoin, is a very risky way to do business.