What are the appropriate usages in the following?
People vote according to their feelings (or feeling), not necessarily their rationale (or rationales).
People make up their minds (or mind) slowly.
People do not necessarily follow their logic (or logics) when making decisions.
I would use feelings and minds in the plural in your sentences, as each person has their own mind and feelings. Only use them as singular if all the people share the same mind or feeling, such as when talking about a board of directors: They made up their (collective) mind to abandon the project.
They went with their (common) feeling that John was the best candidate.
Rationale and logic are usually used in the singular form as they describe a general, abstract notion, not specific things, e.g. What was their rationale/logic (=way of thinking, reasoning) in making this decision?