When should the word "English" be capitalized?

Solution 1:

If it is a proper noun, it must be capitalized.

If it is an adjective derived from a proper noun, it should retain its capitalization, according to this Wikipedia entry:

In English, adjectives derived from proper nouns (except the names of characters in fictional works) usually retain their capitalization
– e.g. a Christian church, Canadian whisky, a Shakespearean sonnet, but not a quixotic mission, malapropism, holmesian nor pecksniffian.

Where the original capital is no longer at the beginning of the word, usage varies: anti-Christian, but Presocratic or Pre-Socratic or presocratic (not preSocratic).

The "usually" might explain why you sometimes see "english" without any capitalization. The only case of "english" as a common noun would be in the context of pool, billiards or bowling games, as described by Wiktionary:

english (uncountable)

  1. (US) Spinning or rotary motion given to a ball around the vertical axis, as in billiards or bowling.

You can't hit it directly, but maybe if you give it some english.

Solution 2:

Unless you mean spin on a billiards ball, it should be capitalized.

Solution 3:

I often confused how the word "English" should be written in phrases like "English language", because I meet both variants: capitalized and starting with lowercase letter.

Hmm, really? Probably just seeing mistakes.

What is the most accepted usage: "English language" or "english language"? And what about other possible usage of the adjective "english"?

"English language" with a capital.

There are some uses of national adjectives which don't have to be capitalized, such as "french windows". Presumably the same logic would apply to something like "english muffins". However I don't have statistics as to which is more common.