What is supervenience and is there controversy over its definition?
Does it have to do with the list of qualities of two 'things' being compared?
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I spent 6 years as the music director at a Lutheran church. The concept of supervenience is quite controversial with conservative Christians (and likely other similar groups) because it implies that every thing we do, think, feel and believe is derived in organic chemistry. And thus the house of cards plummets. I actually reject this kind of false binary logic because God (assuming God exists) could have set the natural laws of the universe in motion, allowing chemistry & biology to run its course. It's amazing to me how some Christians claim God is all powerful and all knowing, but seem to deny that God could have done it this way. Anyway...just some food for thought I guess.
I don't think there is much controversy in its definition. Quoting SEP, "A set of properties A supervenes upon another set B just in case no two things can differ with respect to A-properties without also differing with respect to their B-properties." The most common example to motivate such a definition is to think of an image and the pixels that compose it. The image supervenes on the properties of the pixels since the properties of the image completely fix the properties of the image. A more philosophical example might be the claim that all biological truths supervene on the totality of all physical truths. More controversially, we have the claim in the philosophy of mind that all truths about phenomenal experience supervene on physical truths. Claims about zombies will be relevant here.
Looking at the comments, the worry about multiple realizability is just that, for example, the mental fact, "I am thinking about Paris" is multiply realizable by many, many different physical states of my brain. So, at a first glance, you might think it is problematic to identify thoughts of Paris with a particular brain state. The natural thing to do is to look at types rather than tokens, however. This example just shows that the supervenience relation doesn't go the other way. It's not the case that having a thought about Paris fixes what brain state you are in. However, most would say your physical state does fix what mental state you are in.