Learn English – When should “doctor” be capitalized

This Writer's Block page on capitalization sums up the rules in one page which is the most useful that I have found, basically these rules from the Chicago Manual of Style plus a number of minor rules which are worth reading:

1. Always capitalize the first and the last word.
2. Capitalize all nouns, pronouns, adjectives, verbs, adverbs, and subordinate conjunctions ("as", "because", "although").
3. Lowercase all articles, coordinate conjunctions ("and", "or", "nor"), and prepositions regardless of length, when they are other than the first or last word. (Note: NIVA prefers to capitalize prepositions of five characters or more ("after", "among", "between").)
4. Lowercase the "to" in an infinitive.

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.