Bill likes Sarah's singing jazz ballads.
Sarah sings jazz ballads. Bill may hate the ballads themselves, but he likes that she sings them.
Bill likes Sarah's singing solo.
- Bill likes the solo that Sarah sings. Singing is an adjective modifying solo.
- Bill is happy that when Sarah sings, she is singing solo. Singing is a gerund, modified by solo.
Bill likes Sarah's beautiful singing jazz ballads.
For either verb or noun form uses of singing these are both wrong. Singing can be an (informal) adjective, however: Singing (music) Smooth and flowing. (http://en.wiktionary.org/wiki/singing#Adjective)
Example: Brian Sanborn's playing transcends guitar with his singing melodies and cloud-like chords. https://www.google.com/search?q=%22his+singing+melodies%22&ie=utf-8&oe=utf-8
With that definition, the sentence parses fine - Bill likes Sarah's ballads. Ballads is the object of the sentence, and is modified by two adjectives, beautiful and singing.
Idiomatically, however, it is a little awkward, and might not mean what you might think it means. Singing as an adjective meaning smooth is often used to modify instrumental play or melodic structure. Singing piano melodies are piano melodies that sounds like humans singing. For obvious reasons, it is not used to describe singing itself. (Her singing singing sounds just like a person singing?)
If Bill likes Sarah's singing ballads means something, it is that she is a trumpet* player whose ballads sound smooth and flowing - like singing - and Bill likes them.
Bill likes Sarah's beautiful singing solo.
Two possibilities here. Singing is an adjective either way.
Bill likes the solo that Sarah sings.
She is a trumpet player whose solos sound smooth and flowing - like singing - and Bill likes them.
Finally - I agree with what TRomano said in another answer. These are all very awkward constructions, and even though I can make grammatical sense of them, they don't sound natural.