Government owned and operated (Noting the difference between Government and State)
State owned enterprises
Public private partnership
Privately owned( Limited company, partnership, sole owner etc)

Does Privatisation work?

It is important to note first that it is just as easy for the Government/Private Sector to make a complete mess of running enterprises. Companies can always be run better, no manager of a company/enterprise is perfect. 
The success of privatisation depends a lot on how its done. ie. Making an enterprise private, but keeping the legal monopoly will not add the benefits of competition.

Of the options above..
Option 1) 

This is one of the worst options because the operation is interfered with by the government of the day and so not always in the interests of the operation. Also the government will give the operation preferential legal treatment for these operations at the expense of normal citizens. 
The enterprise run by the government becomes a political football, and when things go wrong the taxpayer bails them out.
Also there tends to be a centralised management which means if you want to change something like the enterprises opening hours, you need to change the whole government to do so.

Option 2) 

This tends to be a lot better than option 1, but it is still prone to the problems of its owner. Governments might tell the SOE to give it dividends rather than invest to get better services. 
When SOEs need more money they have to ask at a time when its politically not unpopular.
You quite often see political appointments running the company(former MP's) instead of  qualified business leaders.

Option 3)

 This is probably the worst option because instead of the best form both the private and public, it tends to bring the worst from both. These enterprises are not usually accountable to the public. (Govt blames their partner and business blames Govt when it goes wrong).

Option 4) 

Private business works all the time, its just a bit haphazard the way it goes about it. Think about all the things you get from the private sector at low prices like: eye laser surgery, food, clothes. IT has been constantly demonstrated in countries around the world, when the government provides these, there tend to be long lines and rationing.
Private businesses go bankrupt and make mistakes all the time, this is how bad management and obsolete services are removed from the system. A process of good management discovery.(Darwinism in business) . This is what SOE's miss. When something goes wrong, they don't go bankrupt, they tend to turn into sink-holes where money keeps getting pored into to keep afloat. 

If you’re against state-run trains, you’re against trains, if you’re against state-run hospitals, you’re against hospitals and so on. This is clearly a fallacy or misrepresentation

If charging nothing at the point of use is the apotheosis of a good service, then socialists could be right. But, of course, nothing is free, since it is ultimately paid for by someone somewhere. If, however, a good service is about offering the best quality for the lowest price, then the free market wins over time.

Essentially the Left hates to think that the type of person who is good at making money could be good at anything else - particularly anything to do with enriching the soul, satisfying our aesthetic tastes or increasing happiness. They prefer to think that this is the strength of clever, compassionate people like them, rather than the cocksure business types who get their backs up.

The basis for this is that government has the power to tax and it should not use that to compete with those whom it taxes.

When the public are asked if they want to have state ownership, they should always be asked if they want it even if quality goes down.

^Link276 the other side of the argument

^Link278 privatisation of railways