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Many german words, in plural form, take an umlaut. For example:

Mutter -> Mütter

The dictionary I'm using, uses a form of abbreviation like this:

s Rad,-: er tyre

This gives me the following information:

form | singular | plural
word |  das Rad | die Räder

I'm specifically interested in the two dots before the er as they signify that the vowel will change to an umlaut.

However, the problem is when the word at hand has two vowel. Example:

r Ausdruck, : e Expression

In this case there are two potential candidates for the umlaut. (in this case it was the second one).

Here is another example:

Wasserball, :e water ball

My initial hunch was that the last potential candidate in the word gets the umlaut. However, I stumbled upon a counterexample:

Kaufhaus, - : er department store

In this case the a gets the umlaut.

Is there any rule in which one can know which letter in a word gets an umlaut in its plural form?

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    The German word Mutter has a second meaning: Mutter is also the German word for the nut with an internal (female) thread that you can screw onto a bolt with an external (male) thread (which is Schraube or Bolzen in German). This technical Mutter has a different plural than the female parent, and it has not an umlaut. It is die Muttern: »Man muss acht Muttern abschrauben um den Deckel zu entfernen.« = "You have to unscrew eight nuts to remove the lid." – Hubert Schölnast Sep 30 '18 at 4:09
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    A similar case is your example Ausdruck which also can mean print/printout. If it is used with this meaning, the plural would be Ausdrucke which has no umlaut in the plural form. So context is key in some cases. – tallistroan Sep 30 '18 at 6:29
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First of all, those two dots are the vestiges of a medium-old-fashioned way to write the letter e on top of a vowel —pretty similar as Swedish and Danish do it with the little o on top of the letter å— In much older German texts, you will actually find a small e printed on top of another vowel.

And that pretty much explains what Umlaute are. They are a special kind of Ablaut, involving the vowels a, o , u and the vowel e.

das Kaufhaus, die Kaufhäuser

The letter combination au is a Dipthong which sounds similar to the sound English writes as ow. It's ablauted partner is äu, and that's it. German has some more Dipthonge, but au/äu is the only one involving an Umlaut. Äu sounds just the same as eu (English writes this sound oi or oy.)

That's the one exceptional case. The general rule for doing the Ablaut (for plural, past tense, downsizing, etc.), however, is

  • put the Ablaut on the last non-reduction syllable of the last component, ignoring suffixes and case/tense endings.

Often enough, this is just the first syllable of the last component, ignoring prefixes. This explains why it's

Es ist sechs Uhr morgens. Die Kaufhäuser sind noch geschlossen.

Auf dem Wochenmarkt wird schon gekauft und verkauft.

Käufer und Verkäufer feilschen um die Preise.

An ihrem Verkaufsstand bieten die Obstverkäuferinnen auch Gemüse an.

An den anderen Verkaufsständen gibt es Wurst und Käse.

The last component of Obstverkäuferinnen is Verkäuferinnen, the plural nominative ending is -nen, the female occupation suffix is -in-, the actor suffix is -er-. The syllable to put the Ablaut on is kauf. And as that's the au Dipthong, it gets äu.

Important note: This rule only explains where to put the Ablaut, but not if it has to be put there or not. The latter depends on a lot of factors. As a language learner, you better stick to learning

  • the gender (always recap the nominative singular der/die/das along each word)
  • the nominative singular
  • the genitive singular
  • the nominative plural

This is what a dictionary gives you for each noun. It's the only information you need to put the noun into one of the patterns which you will learn automatically as soon you are sufficiently exposed to German.

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There are quite a lot of misconceptions in your mind and in your questions, Mhmd. Janka discussed most of them but forgot the worst one.

"My initial hunch was that the last potential candidate in the word gets the umlaut. However, I stumbled upon a counterexample: Kaufhaus, - : er department store"

There are no "candidates", but if you think that this expression helps YOU, okay. Thus in Kaufhaus the "candidate" is -haus, not the vowel -u-. (your allegedly "last potential candidate") In other words: the "candidate" is not a singular vowel, but a whole word. Ergo: das Haus -> die Häuser. Which is part of the elementary vocabulary, so where's the problem?

2nd misconception: Umlaut-dots are not confetti that one could throw just anywhere.

There are 3 monophthongs that can take an umlaut: -a-, -o-, -u-

der Ball -> die Bälle, die Stadt -> die Städte / der Chor -> die Chöre, der Hof -> die Höfe / der Hut -> die Hüte, die Nuss -> die Nüsse

There is ONE dipgthong that can take an umlaut: -au-

das Haus -> die Häuser, das Maul -> die Mäuler, der Traum -> die Träume

-u- as a "candidate" is part of your Träume. ;-)

Cheers!

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I shall try to explain why certain vowels undergo umlaut and others don't.

It appears to have been a universal feature of Proto-Indo-European (PIE) that the vowel in the ending of a noun always changed to an e or i - or a diphthong including one of these - in the nominative plural (masculine and feminine, but never neuter) and often in the genitive singular (all genders). We see this in virtually every plural form in Latin and Greek:

cactus -> cacti larva -> larvae

Umlaut in Wikipedia gives an example of

*fōts -> *fōtiz in Proto-Germanic

This is often called palatalization as the tongue tip goes up towards the palate to make these vowels, but terminology varies. They are often called high front vowels. Sometimes it is just a front vowel, without the tongue being that high.

You still see this change in modern Italian:

americano -> americani pizza -> pizze

This shows why you don't get an umlaut on an e or an i or any diphthong containing one of these: the tongue has already moved, and can't move any further. But this is in the endings - how does the vowel in the stem get affected?

It appears that speakers of the Germanic and Celtic languages simply started moving their tongue too soon, before they had got to the end of the word, so the last vowel of the stem underwent umlaut instead of the ending. So

*fōts -> *fōtiz became

Fuß -> Füße the e sometimes survives in the ending as well

foot -> feet

There is yet one more problem. If you try to make these sounds and see where your tongue goes you will find that the vowels in English mice and German Mäuse may not be front vowels at all, depending on your dialect. But this is simply that the sounds have changed after the grammar was established.

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