In chapter 30 of Wuthering Heights the relationship between Zilla and Cathy is mutually contemptuous, with both parties having perhaps genuine reasons to dislike one another.
Nelly explains that Zillah, her only source of information from the household, is "narrow minded and selfish." She also says that the relationship between her informant and Catherine deteriorated since the first time the young Cathy came into the household.
According to what Zillah tells Nelly about that first incident, Catherine was in need of help from Zillah, but the latter did not do anything to help because she was instructed by Healthcliff to let Cathy fend for herself. Resentful about this treatment, Cathy considered Zillah her enemy, and began to treat her as a rival.
Catherine evinced a child’s annoyance at this neglect; repaid it with contempt, and thus enlisted my informant among her enemies, as securely as if she had done her some great wrong.
Zillah's version of the same issue is that she does not like Cathy because she feels that the girl is haughty and spoiled. Evidently the silent treatment that Cathy gives Zillah really annoys the woman. This is why Zillah tells Nelly that while Cathy may be of a better station and origin, she is not "too good" for Hareton.
...you happen think your young lady too fine for Mr. Hareton; ...but I own I should love well to bring her pride a peg lower...what will all her learning and her daintiness do for her, now? She’s as poor as you or I: poorer, I’ll be bound: you’re saying, and I’m doing my little all that road...
When Cathy was in mourning, Zillah tried to show Cathy a degree of care and sympathy. However, Cathy rejected every one of these attempts. This basically adds more negative emotion to a relationship that is already jaded.
Therefore, Zillah and Cathy are mutually mean to one another. However, since Zillah is the servant and her master has asked her to not help Catherine, her resentment against her is all the more evident.