Some nouns, like "clothes," exist in the plural only.
Is there a list of such nouns, or a rule or pattern that describes the set, and can be used to figure out if a singular form of a plural noun does or does not exist?
They're called pluralia tantum (Latin for "plural only", singular form: plurale tantum), defined by Wikipedia there as...
a noun that appears only in the plural form and does not have a singular variant for referring to a single object. In a less strict usage of the term, it can also refer to nouns whose singular form is rarely used.
There's a list of a dozen examples at oxforddictionaries (trousers, scissors, glasses, clothes, marginalia, folk, shenanigans, loggerheads, cahoots, amends, smithereens, thanks) from which it should be obvious that some are only pluralia tantum in specific contexts with specific meanings.
I don't think there's any "rule" that would help you identify which words work like this (or in which contexts). You just have to remember it as and when you learn each relevant word and meaning.