Where do the tone marks go?

Tone marks in Hanyu Pinyin always go over vowels, not consonants. But even those familiar with Hanyu Pinyin are often uncertain about which in a string of vowels takes the tone mark. If, for example, you are given "huai4" -- is it hùai, huài, or huaì? (Answer: the second choice.)
Fortunately there are no ambiguities to worry about, even where there are several vowels in a row. Various complicated rules explain the placement. Fortunately, in application they boil down to a few very simple guidelines:

• a and e trump all other vowels and always take the tone mark. There are no Mandarin syllables in Hanyu Pinyin that contain both a and e.
• In the combination ou, o takes the mark.
• In all other cases, the final vowel takes the mark.

The possible vowel combinations are listed below, with the vowel that receives the tone marked as second tone.

Note: Early versions of Hanyu Pinyin also used ê. But since it never was combined with other vowels it is not included here. (It has since been supplanted by ei.)