Which propositions are considered meaningful and on what grounds? In other words, when is it correct to predicate 'meaningfulness' of the propositions?
The meaning of a sentence seems to me to be a relation between the sentence and something which is beyond the sentence. A senseless sentence is just a sentence, a mere sentence . A meaningful sentence is somehow more than a mere sentence. It is so by being related, in a way that a senseless sentence is not. A meaningful sentence is related to "the external world". The "world" is "external" to the related sentence, it is beyond the sentence.
The above raw definition covers both semantic meaning, where one uses terms like 'refer' or 'denote' to express the relation of words to the world; and pragmatic meaning, where the word-world relation is thought to be practical ("meaning is use").
I agree with one commenter, that all propositions must have meaning. However, the commenter didn't explain why, so I thought I'd elaborate.
A proposition must be true or false. You can't argue either of these if the proposition is "meaningless".
You can redefine propositions more broadly if you like, and consider "meaningful" ones to be those that can be proven or refuted. However, there is no simple way to tell if a proposition is "meaningful" in this case.