Are prefaces common in academic papers? I know that sometimes (in a thesis for instance) one can thank his supervisor, family or whoever and give a small reflection on the work. A good place would be a preface, I think, though I am not sure whether this is common in academic papers. If not, what is a good place to place reflections and - more informally - thank-you notes?
It depends on the type of publication, I would say.

It would not be typical for journal or conference publications to have prefaces since dead trees are at a premium and long prefaces are not useful technical material.

It is common for journal and conference publications to have a short "Acknowledgements" note at the end of the paper (just before the bibliography), or linked as a footnote from the title. This note is typically for acknowledging funding sources of the authors, as well as people who contributed informally to the paper (but are not co-authors). Sometimes I like to thank (even anonymous) reviewers if their feedback was useful. However, these notes are concise: they are not prefaces where you would discuss, for example, the invaluable contribution of your cat or somesuch.

For books, monographs, etc., prefaces are fair game. But do consider what you put in there. It is often the first thing a reader will read.