Transportation and especially aviation projects are often very long term endeavors. They may take 10+ years to come to fruition, if ever. Very likely to be cancelled (in North America) because of our political processes. In Europe and Asia they are probably more secure. The firms working on these tend to be large and often hire and fire large numbers of people during and after downturns. If you can make it through the first 5-10 years they can be stable. 

Healthcare is extremely niche and probably offers the least room for innovation in building program, massing, and organization. Often you are just "designing" meaning making a pre-determined plan look a certain way. Ambulatory facilities tend to be more flexible, intensive care and hospitals less flexible. Projects are more resistant to downturns because people need healthcare even when the economy is bad.

Education is probably the fastest to realize of the three, though still slow by commercial architecture standards. Often when the economy is bad, education projects are still chugging along as the money comes from either appropriations bills and budgets (public) or endowments (private) which are less subject to financing from lenders. They probably offer the most opportunity for innovative and impressive designs. Usually there is a campus architect or city architect you collaborate with and that can be quite rewarding. The program requirements are often open for interpretation and re-thinking. 

All of that said, especially if you are early in your career, I would encourage you to choose the firm that has the most diverse practice, and also the one that does the most of their own Construction Administration. There is no better way to learn than working on different project types and going to the job sites - both in terms of practical and design skills, and the process of learning what you will actually enjoy.

I have interviewed across all types of such firms. In my experience, the answer to your question depends on the context.

What role are you interviewing for? What is your location and do the firms take on local or international work? 

When it comes to transportation is that an architectural firm or A&E firm like Arcadis/Atkins? In aviation projects, a starchitect teams up with an executive architect with the exception of smaller terminal works. Other transportation project types like rail involve mostly coordination of engineering works rather than architectural design. 

Education is indeed the most profitable with repeat business however there is a distinction between higher education and schools. The first is where higher design quality is found whereas the latter are usually lower quality, bread and butter projects with quick project cycles where technical design and CA may be passed on to the contractor. Also the client liaison with such institutions can be unecessarily complex.

With respect to healthcare, there is demand but funding can differ per area of practice and respective policy. For instance, the NHS in the UK has much lower budgets for infrastructure compared to Nordic countries. Also, I have heard that transitioning from healthcare to other practice sectors can be difficult. However, practice areas such as lab design are recession proof. 

Finally, what is your motivation for moving towards public sector work? What kind of firms have you previously work for, and are you prepared to work at places with a more corporate culture like the ones usually practicing in the above sectors? 

I'm interviewing for mid-level positions. They are all large AE firms. I'm in a major city on the west coast of US.