White spot disease

While not particurly prone to this ailment, the disease is common enough that it's worth mentioning. Easily cured.

Oodinium

"Velvet" is probably the most common scourge of killies, although only Nothobranchius seem to be particularly prone to it. Relatively easily cured.

Fish tuberculosis

A pathogen giving fish tuberculosis-like symptoms it has no cure and can be tranmitted to humans. About 20 cases in 30 years are known. It can be cured in humans, but not fish.