The context of the sentence in relation to the other sentences in a paragraph should guide the decision to use a singular or plural noun.
I can't offer any simple solution
I can't offer any simple solutions
Either one is correct. The better choice depends upon the greater context. The second may sound more correct because "any" as an adjective can be defined as "one, some, or all indiscriminately of whatever quantity" (Merriam-Webster) which implies that the object being modified is plural. Another definition of "any" is "one or some indiscriminately of whatever kind" which does not imply plurality.
I want to get better school results
I want to get a better school result
Again either is correct.
My homework and test grades stink. I want to getter better school results.
I did poorly in that class at summer camp. I want to get a better school result
In the first, "results" is referring to school grades. In the second, result refers to a specific class.
They can't obtain a sense of accomplishment
They can't obtain senses of accomplishment
Technically, both are correct.
They can't obtain a sense of accomplishment. The ladies have to create a goal for the committee first.
They can't obtain senses of accomplishment. The ladies have to create their individual goals first.
In the first example, the ladies share a single "sense" as indicated by their participation in a committee. In the second each lady has her own "sense" as indicated by "individual". That said, just because a sentence isn't technically incorrect doesn't mean it might not sound off. In my ear that is a result of the combination of "senses" (plural) and "accomplishment" (singular).
Students should do more sports to help them build a healthy body
Students should do more sports to help them build healthy bodies
The context of sports and their effect on health indicate that each student has his or her own body and so the second is correct. As you mention, the students aren't building Frakenstein during gym class.
June Casa Grande's Grammar Snobs Are Great Big Meanies has a great chapter on singular and plural word choices.
The Rolling Stones are awesome.
The Who is awesome too.