Your research statement (not "statement of purpose" as university administrators strangely insist on calling it) should be a description of your research experience, interests, and goals, as a supporting case for your potential for independent research. If you were applying to work in my research area, I would much rather read about your research ideas than mine.
Here's what I like to see in research statements:
What have you already done? What problems have you solved, or at least worked on? What independent projects have you been part of? What were your key contributions? What did you learn? What did you teach the world? How do your results compare to what was already known? What original ideas are you most proud of? Be specific, technical, credible, and confident (but not arrogant). Refer the reader to your web page for more details. Have a web page with more details: preprints, project reports, source code, videos, etc.
What are you working on now? What problems are you currently trying to solve? What are you currently trying to build? What are the open research issues? What intellectual tools are you using, or learning to use? What are you reading? What prior results are you building on? How are you building on your own earlier work? What is your current favorite half-baked idea? Be specific, technical, credible, and confident (but not arrogant). If you have partial results, refer the reader to your web page for more details. Have a web page with more details.
You ask whether you should worry about the feasibility of your ideas. No more or less than if you were actually doing research. Ideally, I'd like to see both that you have good ideas and that you're not afraid to have bad ones.
- What might you want to work on in the future? What problems you would like to solve? What would you like to build? Do you want to push your existing projects further? If so, how? What new areas are you interested in exploring? What are your long-term career goals? Convince the reader that you are knowledgeable about your target area(s), but be honest about your ignorance. Be specific, credible, and confident (but not arrogant).
No one will hold you to your stated plans. You don't need to do an extensive literature survey (although it certainly can't hurt; put a copy on your web page), but at a minimum you should have had a strong undergraduate course on the topic.
- How does my department fit your research goals? (If the rest of your statement is well-written, the reader already knows the answer to this question, but you also need convince the reader that you know.)
A few sentences about motivation are worthwhile, but don't talk about how many programming languages you know, or about how you were a child prodigy, or how many contests you've won, or how many A+s you got, or how Alan Turing was your childhood hero, or how your research area will Change The World. Write in simple, direct, flawless English. Like Suresh, I have seen many phrases like "my first trembling steps", but never in successful applications.
The only credible reason to "target a particular professor" is if that professor's research interests already mirror your own. If you try to craft the bulk of your statements to different professors in different departments, the result will be much shallower, and therefore much less persuasive, than if you describe your own well-developed research interests.
Yes, that means you have to have well-developed research interests. That's the point.