How to attack a general polyalphabetic cipher? I am able to decrypt Vigenere ciphertext using the index of coincidence and chi-squared test.

However, out of interest, how do you go about attacking ciphertext that was encrypted using a mixed alphabet shifted 26 times?

Also, what about a ciphertext that has been encrypted with 26 random alphabets? (so each line in the tableau is random)

Googling seems to bring up the basic Vigenere, but either an example or a link to a more detailed resource would be good.
A general polyalphabetical cipher is just a combination of several general monoalphabetical ciphers, each applied on every $n$-th letter of the message.

So the first thing is to find out what $n$ is (i.e. the key length). For this we can use the index of coincidence just like for Vigenere.

Then we can split the message into $n$ parts (columns), and try to break each of it as an individual monoalphabetic cipher, starting with frequency analysis. (We'll have to correlate information from the individual columns to do bigram frequency analysis, too.)