The singular form in genitive case of »die Frau« is »der Frau«. You can look this up in any conjugation table, for example, at Wiktionary. So the correct expression is:
auf der Frau Schoß
Compare:
Das Buch lag auf des Mannes Schoß. (masculine)
Das Buch lag auf des Kindes Schoß. (neuter)
Das Buch lag auf der Frau Schoß. (feminine)
What is marked bold above is a left genitive attribute inside a dative object, which, in turn, is marked italics (which itself is inside a prepositional object, which is not marked, it is »auf« + the dative object). Of corse, you can use dative objects also without left genitive attributes, but then you have to use another determiner instead:
Das Buch lag auf dem Schoß. (definite article)
Das Buch lag auf einem Schoß. (indefinite article)
Das Buch lag auf ihrem Schoß. (personal pronoun)
But you also can use right genitive attributes in nominal groups, but then you still need a definite article on the left side of the noun:
Das Buch lag auf dem Schoß des Mannes. (definite article + right masculine attribute)
Das Buch lag auf dem Schoß des Kindes. (definite article + right neuter attribute)
Das Buch lag auf dem Schoß der Frau. (definite article + right feminine attribute)
auf Mutters Schoß
This is more complicated, because this is not the normal behaviour of the word »Mutter«, which has no genitive-s. When you look at Wiktionary, you will find:
genitive of die Mutter is der Mutter (without s at the end)
So, here we have an exception. The standard form of »die Mutter« in genitive case is »der Mutter«, for example, when you use it as a genitive object or a right genitive attribute:
Die Kinder standen traurig am Grab und gedachten der Mutter. (genitive object)
Die Kinder standen traurig am Grab der Mutter. (right genitive attribute inside a dative object inside a prepositional object)
(note: am = an dem = preposition + article)
But »Mutter« will get an extra s at the end when used as left genitive attribute, but only if used without an article:
Die Kinder standen traurig an Mutters Grab. (left genitive attribute without article)
Die Kinder standen traurig an der Mutter Grab. (left genitive attribute with article)
This is true for all terms for female family relationships when used as left genitive attribute without its article:
Ich fahre mit dem Auto der Schwester. (right attribute)
Ich fahre mit der Schwester Auto. (left attribute with article)
Ich fahre mit Schwesters Auto. (left attribute without article)
This works also with Tante, Oma, Cousine, Nichte and of course also with Mutter.
Masculine relatives always have an s at the end in genitive case:
Ich fahre mit dem Auto des Vaters. (right attribute)
Ich fahre mit des Vaters Auto. (left attribute with article)
Ich fahre mit Vaters Auto. (left attribute without article)
So, the genitive s is always there when you use terms for masculine relatives, i.e., the s for left genitive attributes without article is not an exception.
About usage:
Left genitive attributes become more and more outdated. They are still correct, but the standard usage is the right genitive attribute:
Ich nehme die Tabletten auf meines Arztes Empfehlung. (correct, but outdated)
Ich nehme die Tabletten auf Empfehlung meines Arztes. (modern standard)
Other questions, not answered here:
- »Futtern wie bei Muttern«
You already linked to the right question in your posting, so you already know that this is something completely different (it is a regional dialect).
- Fugenlaut
Also epentheses are something completely different. When you search for "Fugenlaut" you will find lots of questions dealing with this topic.
