Mechanical engineering is a great field of study. I myself have a degree in ME, but I work in the aerospace field. A relative of mine works in the oil industry and he has a degree in mechanical as well. Old classmates of mine that graduated in ME currently work in more computer science-related fields (CAx tools). Having an ME degree tells employers that you are versatile.
Aeronautical engineers, from what I've been told, have a very similar course of study to mechanical engineers. The difference comes in the last few semesters of study, where they take more classes in aircraft structures, composites, aerodynamics, advanced thermodynamics and heat transfer, stress analysis, and advanced fluids. It's all basically classes that deal with phenomena specific to aircraft and there's enough classes that differ from the ME degree to warrant a different title. Aeronautical engineers usually work for aerospace companies, but it doesn't mean that they can't do other ME work. I would imagine that the AE degree gives you an advantage on your resume if you're applying for work an aerospace company, but it might not be that advantageous if you were applying, let's say, in the energy field.
About the stability of jobs, I agree with hoomanya. I know that during the "Great Recession" the aerospace industry tanked. However, from what I've been told, it's been a great industry to work in when the economy was booming. Like most jobs, it depends on the economy. However, I do believe that engineering professions in general are inherently more stable than most other kinds of jobs. About specialization, you might consider doing a degree in mechanical and then, if you find out what you want to specialize in, do a graduate degree in that field. You could do ME first, then get a masters in AE. You could even do a masters in a (mostly) unrelated field. I've seen MEs that graduated and really loved computer science, so they got a masters in CS. They took a couple years longer than other CS grad students because they lacked the undergrad basics, but it's not impossible. Some schools allow you to get two masters degrees simultaneously. An ME prof I know got two masters, one in EE and the other in ME (he works doing MEMS, so both have helped him). You've definitely got options, so don't be afraid to choose something.
