If you're really uncertain, always use -ise. There are only about four words where -ize is obligatory, viz prize (= reward) and size, assize, and capsize. Now to business.

The -ise ending in English comes from various places. Where it's part of a Latin verb root, it has to be -ise. (Latin had no Z.) These roots are

In almost all these the bit before the root is a familiar Latin (or French) element like a preposition: circum- 'around', ex 'out', com-pro- 'with' + 'before', enter- 'between', sur- 'on', etc.

Then there's rise (and thus arise). The few other -ise-only words listed above (advertise, merchandise, franchise, chastise) are an odd bunch where no good reason can be given for the retention of S. (Or it's buried too deeply in Latin and/or Middle French to be worth thinking about.) These just have to be memorized. Z is occasionally used, but this is not the official spelling even in American style.

The doubt only arises with the EYES pronunciation. When it's something else (such as ISS as in practise or EEZ as in expertise), it's -ise rather than -ize. (Ignoring the fact that there's also a noun practice.)

Apart from these, the suffix is almost always from the Greek -ιζω or -izô, which was borrowed into Latin (though but rarely) as -izo (okay, Latin did have a Z when it borrowed it from Greek).

This is the living suffix, the one that can be freely attached to create new words: mesmerize, vulcanize, GUI-ize, etc. etc.

Historically, in English both -ize and -ise have been used for these. In Britain but not America, French influence (always -ise) preponderated, and most people apart from Americans use the -ise spelling in handwriting. The -ise is also used in newspapers and magazines throughout the British-language domain, if I may so call it.

In books however, it is different. British printed books from all (reputable) publishers always use -ize (in those words that allow both). This practice arose around 1900, as far as I can tell, under the auspices of Fowler and the OED. While the generic IANA tag for British English is en-GB, there is a specific one for Oxford dictionary style, en-GB-oed, which ensures that -ize is recognized as a British spelling.

This might surprise quite a few Britons who think the Z is an Americanism: no it's not. The Z is used by Oxford, Cambridge, Faber, Abacus, Picador, Penguin, Vintage, the lot. You have to go really downmarket to find S used. But although all publishing houses have adopted it, it has not passed into popular use; and the BBC and the Government still use S.

Oddly, Australian books do use S, and official government style guides in Australia stick to -ise.

The reason why analyse, catalyse, paralyse are still written with S in British books is that they are actually malformed Greek as they stand: the combination of Greek roots would give ana-lys-ize. One syllable has been dropped (haplology) and a compromise spelling adopted.