I've started to learn grammar online (http://uastudent.com/category/english/english-grammar/) and here is what I've learned so far about suppositional mood.
Analytical form, built up with the help of the auxiliary verb "should" for all persons + infinitive. Expresses problematic actions (desirable, suggested, required…), which are not necessarily contradict reality. Expresses the same modality as Subjunctive I, that's why they can be used in the same syntactic structure. They are quite interchangeable and differ mainly stylistically.
In simple sentences. Used only in interrogative sentences beginning with "What if…" (And what if he should come back?).
In complex sentences. In Nominal Clauses it is used in Subject Clauses beginning with the introductory "It" in the principal clause when the modal meaning is expressed (It's important that he should come on time), after the expression of emotion (It shocked me that he should have been so cruel) and after the expression of fear (I feared lest he should find it out). In Object Clause it expresses the meaning of recommendation, suggestion, advice, insistence, etc. (He ordered that we should come). It may be used in Attributive and Predicative clauses. In Appositive clauses it usually modifies abstract nouns (idea, feeling…) and introduced by conjunctions "that, whether, as if, as though" or adverbs "how, why". The parts are not separated by comma (Her feeling that something else should happen surprised me; There's no reason why you shouldn't read it).
In Adverbial Clauses of purpose the Suppositional Mood is introduced by the conjunctions "so that, in case, lest" (He whispered these words lest somebody should overhear him). In Adverbial Clauses of concession it is introduced by the conjunctions "though, although, whatever, however, etc." (Though he should make every effort, he can't make it to the top). In Adverbial Clauses of condition referring to the future it's used to show that the action is possible though unlikely. In the principle clause the Imperative, Conditional or Future Indicative is used (Should he drop in tell him to wait for me; If I should meet her, I should tell her all about it; Should it be snowing we will stay in).
