I want to learn the different shades of meaning and frequency of use of the following verbs that have the meaning of to die:
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As suggested, here an extensive list, sorted by tone and kinds of death. There are several entries in the thesaurus which I am not familiar with – feel free to add them.
Kind of death:
Everything unmentioned is general purpose, with exception to self-inflicted deaths. In formal context (funeral, death announcement, etc.) all formal verbs turn to general purpose.
Only the following are applicable to self-inflicted deaths, e.g., suicide or Darwin Award candidate:
davongehen, uns verlassen, von uns gehen, versterben, sein Leben lassen, sterben, umkommen, ums Leben kommen, den Löffel abgeben, abkacken, den Weg allen Fleisches gehen, über den Jordan gehen, über die Wupper gehen, aus der Welt gehen/scheiden, die ewige Ruhe finden, sein Dasein/das Leben vollenden, in die ewigen Jagdgründe eingehen
If you refer to the death of a person in a formal context, like in an encyclopedia or newspaper article, you would most commonly use 'sterben' or 'versterben'. 'ableben' has the same neutral meaning, but it's far less common and rarely used in its verb form.
Animal deaths are usually referred to with 'verenden', but if it's a pet or otherwise respected/beloved animal, you would use 'sterben'.
Deaths of plants are always referred to as 'eingehen'.
sterben is neutral and can be used for every kind of death. Also this is by far the most used.
ableben is a little bit elevated or euphemistic. It is usually used for quiet and slow deaths, i.e., by illness or old age. For example, this would usually not be used for somebody killed by an explosion.
versterben – if used in inofficial context: same as ableben, except for the euphemistic touch. If used in official context: same as sterben.
abtreten actually means to resign, to surrender one’s post or to exit. It is only used colloquially in the context of dying and at least with a grain of irony or metaphor.
erlöschen is only used for dying in combination with Leben. In this case this a very formal style.
Sein Leben erlosch. – He died.
krepieren has a disrespective note and also implies a painful or miserable death.
Krepier, du Schwein! – Die, you bastard!
verenden is usually only used for animals. If used for humans, this implies disrespect for the dying person.
eingehen is almost only used for plants and lower animals, where it implicates a slow death. I have never seen it used for humans.