What is the correct plural of "octopus"?

What is the correct plural of octopus:

• Octopi?
• Octopodes?
• Octopuses?
• Something else?

I would go with octopuses.

That is part of the Wikipedia "Plural form of words ending in -us" article:

Currently:

• octopuses is the most common form in the UK as well as the US;
• octopodes is rare,
• and octopi is often objectionable.

The Oxford English Dictionary lists octopuses, octopi and octopodes (in that order);
it labels octopodes "rare", and notes that octopi derives from the mistaken assumption that octōpūs is a second declension Latin noun, which it is not. Rather, it is (Latinized) Ancient Greek, from oktṓpous (ὀκτώπους), gender masculine, whose plural is oktṓpodes (ὀκτώποδες).
If the word were native to Latin, it would be octōpēs ('eight-foot') and the plural octōpedes, analogous to centipedes and mīllipedes, as the plural form of pēs ('foot') is pedes.
In modern Greek, it is called khtapódi (χταπόδι), gender neuter, with plural form khtapódia (χταπόδια).

As other answers have already pointed out, the correct English plural is octopuses.

Some clarification is needed in the answers, though, since many people are under the impression that octopi is definitely wrong for historical reasons, while octopodes is apparently acceptable (even though almost no one aside from grammar fanatics has ever heard of it).

After reviewing the historical evidence, both octopi and octopodes have problems.

Aside from the fact that octopi is a familiar plural to many and is indeed listed ahead of octopodes in almost all (if not all) dictionaries, generally following octopuses, there are strong historical reasons why octopodes should be suspect.

1. The word octopus did not actually exist in either ancient Greek or ancient Latin, so appeal to a "native" plural is problematic. The standard word for the Romans and Greeks to refer to the animal was polypus/polypous. (There was a very rare adjective in Greek oktapous, also incredibly rare in Latin form as octipes, but it merely meant "eight-footed" and did not refer to the creature we now call an octopus. It's exceedingly unlikely that our modern word came from these roots, as the etymology is explained below.)
2. While the root octopod- is suggested by English words like octopod and the biological order Octopoda, there are also English words drawn from the root octop-, such as octopean, octopine, and octopic. All of these suggest that there is a significant history of educated folks who would accept octopi as a reasonable plural.
3. The word octopus was coined by Linnaeus (the biologist) in the 18th century, based on a Latinized Greek word polypus. Linnaeus did use the plural octopodes in his modern Latin and the root octopod- for example in the taxonomic order Octopoda. On the other hand, almost as soon as the word entered the English language, the plural octopi begins to be seen. The OED has a sample quotation from 1834 with octopi (only the second English-language quotation in their list), but a cursory search in Google books shows that octopi occurred well before that in the early 1800s. Essentially, there has really never been a period in the English language where octopi didn't co-exist with octopodes. Most sources from before 1850 or so in English are specialized texts listing off Linnaeus's classifications, but once we begin to see the word octopus come into everyday use, octopi seems to become a common, if not dominant plural.
4. "Yes," comes the objection, "but it's still wrong. It doesn't matter how long it's been in the English language. The word comes from Greek, not Latin." First off, the word comes from Latin -- if it were Greek, it would be oktopous (or maybe octopous, if we allow a little fudging in the transliteration). Yes, octopus ultimately comes from Greek roots, but it comes through Latinized Greek. What matters is not how we would decline the word in Greek, but how we would decline the word in Latin. The argument goes that it would be a third declension noun in Latin, with stem octopod-, rather than a second declension noun, with stem octop-. Hence, octopodes over octopi. A difficulty with this argument is that the Latin predecessors, such as polypus, from which octopus was coined, actually used the second-declension plural form polypi. Latinized Greek plurals were often not consistent in their declensions, even for the ancient Romans. Even Linnaeus inconsistently used polypi alongside octopodes, since he knew his Latin well, and the Romans used polypi, not polypodes. Educated ancient Romans, who knew their Greek well, still preferred polypi. Would they object to octopi? I don't know, but when your plural form is actually inconsistent with the plural of the original word used as a basis, the historical argument gets more murky. (I would also suggest that some educated English speakers and writers in the late 1700s and early 1800s who encountered octopus for the first time could have known that the Latin word for the animal was polypus, and they may have assumed octopus would form octopi as a plural as well.)
5. "Okay," comes a final objection, "but Linnaeus knew his Greek, and we do too. If the word ultimately came from Greek, we should use the 'proper' way to pluralize the Latin version, even if the ancient Romans didn't know any better." Ah, but there's a further problem. The ancient Greeks weren't consistent in choosing a declension for polypous. In addition to polypodes, one can easily find examples of polypoi, which would probably be the reason the Romans used polypi.

Is octopodes "wrong"? No, I don't think so. But arguing for an analogy to native Latin or ancient Greek plurals is misguided in this case. For this specific example, the likely form that the Romans would have used (if they coined the word) could have easily been octopi, particularly if they thought of it as related to polypus at all.

The only way that we get to a position where octopodes is definitely "correct" and octopi is definitely "wrong" is by requiring a word coined in the 1700s to follow rules about ancient Latinized Greek plurals that were very inconsistent in ancient Latin, and are explicitly contradicted by similar words in both ancient Latin and Greek in this case.

I personally would avoid both octopi and octopodes and consider them to be sort of "skunked" plurals. If you want to use octopodes to show off some classical skills, just be aware that, etymologically, you're on somewhat shaky ground.