Research statement and dissertation abstract
I have never seen this done, though sometimes people include abstracts to their work on their webpages.  The reason it doesn't make sense to me is that they serve very different functions.  

For abstracts, you want to briefly explain the contents of your paper to experts, or at least people in your field, quickly so they can ascertain whether they want to take a closer look at your paper.

For research statements, you don't want to be terse.  You want to provide background context and motivation for your work, and you're not trying to convince people to read your work, but rather that your work is good and interesting and they should hire you.  Also, you want to convince people you can explain your work well. (Depending on the kind of position you're applying to, your research statement will either be written for non-experts experts.)

Of course, you can say basically the same thing as what you would in an abstract in an introductory paragraph, perhaps being more informal, but I definitely would not set it off as an "Abstract" block inside the statement.