How to choose a recommender during graduate school applications? I am applying for a Ph.D. in Mathematics in U.S. universities from India. The application process requires me to register people who can comment on my capability for research and recommend me for the graduate school. I have heard that since most applicants for (pure) mathematics Ph.D.s generally do not have publications to show, the recommendations are crucial in the selection process. Hence, I have three questions about selecting my recommenders wisely.

1) Do you feel that recommendations coming from a Professor carries more weightage than the ones coming from an Assistant Professor? My question is that, does the seniority of my recommender add any extra weight or credibility to the recommendation?

2) If a senior Professor gives me 'above average'(or 'good') recommendation while another Assistant Professor ( just two years on the job, not very famous but a very active and well cited researcher) gives me 'excellent' recommendation - which one do you think will support my application better?

3) Which recommendation should I prefer - from a person with whom I worked on a project for around a month, or , from someone who has taught me two courses over two semesters?
Choosing recommenders doesn't exactly have clear-cut rules, so it's going to be impossible to give specific advice without detailed knowledge of your exact situation. That said, some guidelines that I found helpful:


Try to cover all your bases. I'd recommend having at least one professor with whom you worked with closely on a research project or something similar, and at least one professor who knows you well from class(es) in which you excelled. Remember that you generally have three recommendations to work with, so make sure all of them are useful.
In general, you want the best letters possible, regardless of rank or prestige. As long as they're faculty and actively do (or even did) research, go for your best letters. Again, this is where you can cover your bases a little with multiple letters, hopefully not all of your potential recommenders are brand-new faculty. 
Don't be afraid to simply schedule a meeting or send an email and ask them for their thoughts. In my experience, most faculty are excited to see their students aspire to graduate school and are very willing to talk about it. You should at least be able to ask your potential recommenders if they can write a good recommendation.
If a recommendation letter doesn't add information to your application, then it probably isn't helpful. If a professor who taught you can pretty much just say the grade you got in the class, then it doesn't add anything over your transcript. Lots of applicants make this mistake, assuming that an A in one or two classes is enough to get a good letter. You can get great letters from professors who only taught you, but make sure that you actually gave them something to talk about in those classes.


I should also say that most people shouldn't have to worry too much about these considerations. It usually comes down to picking the best third letter writer or something. But if you find yourself trying to parse through many decisions in selecting letter writers, then you're probably overthinking it or reaching down too far. Ultimately, developing good potential recommenders is the hard part, and hopefully that has already happened.