When is correct to capitalise these words?

At the beginning of sentence is a given.

Here are some examples for each word. Should any be capitalised? If not, is there any situations where they should?

The word "government"

1. The government raised taxes.
2. Thatcher was in government for a long time.
3. The UK is having problems with its coalition government?

The word "organisation"

1. The organisation needs to make 10% savings.
2. I would like to work for an organisation like Google.

Solution 1:

In most cases you would not capitalize either word on their own.

A possible exception might be if you were to use the word as a proper name, or as an abbreviated version of a proper name. For example, you would use capital letters to name the Organisation for Economic Co-operation and Development; after spelling it out in full the first time, you might shorten the name to "Organisation" on subsequent mentions: "The Organisation is expected to release its study next month." (In this particular case, however, you'd be better off calling it the "OECD," which is its preferred English-language acronym; the same is likely true for many groups with organization/organisation in their names, like OPEC, the WTO, etc.)

For government specifically, in some cases it may be proper to capitalize it as part of a proper name; for example, I believe that Her Majesty's Government (of the UK) and the Government of Canada both like to be referred to as such. Even in those cases, I would probably lowercase the word government on subsequent occurrences when using the word by itself, unless I were dealing with a house style guide that required otherwise.