Can one say "I should like" rather than "I would like"? Is the former grammatical?

My focus here is on the should in the sentence fragment "I should very much like...". Why is it there in place of would? It seems strange that should is used in the subjunctive mood there -- is it grammatical? If the fragment is idiomatic, can anyone explain the history of the idiom?

Great Question! I was intrigued to find the following in NOAD:

As with shall and will, there is confusion about when to use should and would. The traditional rule is that should is used with first person pronouns ( I and we), as in: I said I should be late, and would is used with second and third persons ( you, he, she, it, they), as in: you didn't say you would be late.

In practice, however, would is normally used instead of should in reported speech and conditional clauses: I said I would be late; if we had known, we would have invited her.

In spoken and informal contexts, the issue rarely arises, since the distinction is obscured by the use of the contracted forms I'd, we'd, etc. In modern English, uses of should are dominated by the senses relating to obligation (for which would cannot be substituted), as in: you should go out more often, and for related emphatic uses, as in: you should have seen her face!

Grammatically, once upon a time, will meant want and shall meant something like will.

To be a little clearer saying you "shall" something meant both that you "owed" doing it (or "should" do it in the current sense), and that you were going to do it ("will" in the current sense.) There's a question about shall and will here: When should I use "shall" versus "will"?

As I've noted, will has usurped roughly half of what "shall" used to mean.

Similarly, "would" has usurped the place of "should" in many cases. For whatever reason, "should" has clung to its place in the sentence "I should very much like to..." and in that particular sentence "should" retains its former meaning.

...As to why this particular phrase kept its "should," the answer can probably be summarised as "because it did" and a blank stare. I'll be happy to find I'm wrong, though!

Here's a very small, quick reference attesting to the fact that this use isn't unique, just rather obscure, nowadays:
http://alt-usage-english.org/excerpts/fxshallv.html

Also, an etymology of "should":
http://www.etymonline.com/index.php?term=should

And one for "would":
http://www.etymonline.com/index.php?term=would

It seems to me that "I should very much like..." is more of a British version. You don't seem to hear it much in the States.

Today, this is essentially an artificial prescriptive rule. The (somewhat spurious) argument goes something like this:

• the modals 'will' and 'shall' once upon a time were lexical verbs that carried a notion of 'willing/wanting' and 'command/obligation' respectively;
• "therefore", when used as modals to indicate future tense, that future tense "should" be coloured appropriately with whichever "type of future meaning" (willing vs obligation) is most appropriate;
• and "therefore", the use of 'would' and 'should' "should" also be coloured in a similar way.

For more details, the Fowler brothers dedicate about 20 pages to these arguments in all their glorious spuriosity in "The King's English" and Fowler also dedicates a short amount of space under the 'shall' entry of his Modern English Usage to 'explaining' (although he doesn't really put forward a cohesive argument so much as professes) how the "correct" form is "I should prefer/like etc".

But if you think about it for a moment, the "therefore"s in the above are really non sequiturs. Why on earth "should" the use of a modal auxiliary bear any relation to its use several centuries ago when it was not a modal auxiliary? And why "should" any distinction between 'will' vs 'shall' exactly mimic the distinction between 'would' vs 'should'?

The original premise, that 'will' and 'shall' are originally based on two verbs with distinct meanings, is also not entirely clear. For a start, 'will' appears to be the convergence of two Old English verbs ('willan' and 'willain'). And it's not clear that 'shall' and 'will(an/ain)' had such distinct meanings: even in Old English, it's possible to point to uses where the sense of 'volition' and 'obligation' overlap. (Even 'shall' in its sense of 'owe (money etc)' may imply the notion of 'I will pay you because we have an agreement', i.e. there's a sense of a 'mutual contract' implying both volition and obligation.)

The "rule" appears to have sprung up in the 17th century in works such as Mason's "Grammaire Angloise" (1622). It's not entirely clear whether at the time, the "rule" reflected actual usage, or at least a simplified version of actual usage, or whether it was essentially prescriptive. One thing that the 17th century descriptions have in common is that they present the distinction somewhat succinctly and are possibly oversimplifying for the sake of an educational device. Remarks by Samuel Johnson in his dictionary published in the following century appear to paint a more mixed picture. It's possible that later prescriptivists took the 17th century authors' oversimplification of the situation and assumed them to be more 'absolute' truths. Some studies within the last century have attempted to analyse 17th century usage, notably Fries, 1925 and Hulbert (1947), and reached differing conculsions on the extent to which the "rules" stem from actual usage.