Usage of 'z' in the word serialized in English?

To use an s in specialised is usually the British English spelling. On the other hand, using a z is the American English spelling. You can choose which one to use depending on what your audience is. If you are in Europe, it may make more sense to use the British English spelling.

However, there is no such thing as one single, correct English. There are many, many dialects of English, and even what is known as Standard English varies between countries. If an American sees specialised, they will understand it, just as most British people would understand specialized. Both are versions of Standard English.

As simchona says, there is no single correct English. Every major variety has its own pecularities, and on this particular question at least one major variety -- standard British English -- is itself divided.

In words containing the suffix derived from Greek -izo, the spelling -ize is required in American and Canadian English and not uncommon in British English: the British National Corpus runs 3:2 in favor of -ise, and the Oxford University Press uses -ize. The spelling -ise is much more strongly favored in Australia and New Zealand, but -ize is not unknown.

For further information see this Wikipedia article, this one, and this note by the late Larry Trask.