Appel's research is in software verification, computer security, programming languages and compilers, and technology policy (especially voting machines and election systems); he has published over 150 research articles in those areas. He has worked on fast N-body algorithms (1980s), Standard ML of New Jersey (1990s), Foundational Proof-Carrying Code (2000s), the Verified Software Toolchain (2010s), and formally verified numerical methods (2020s).

Dr. Appel has studied voting machine security and accuracy, election technology, and election procedures since 2004. He has written over 90 blog articles explaining voting-technology issues to the public, as well as 3 law-review and law journal articles on election technology, and taught undergraduate classes on that topic. He has served as an expert witness in voting-machine litigation in New Jersey, Georgia, Alabama, and Arkansas, and assisted state officials defending litigation in New York and New Hampshire. He has testified on voting-machine issues before the U.S. House Subcommittee on Information Technology, the New Jersey legislature, the Pennsylvania Senate, the Georgia State Elections Board, and the New York State Board of Elections. He served on the National Academies of Science, Engineering, and Medicine panel that produced the 2018 report, Securing the Vote. He has served for many years on the Board of Advisers of the Verified Voting Foundation. Since 2025 he serves on the Texas board of voting machine examiners, advising the Secretary of State regarding the certification of voting machines.