As always, I think putting this all down is a good idea and hopefully the anonymity here will allow some students to access the info they need.
I have one thing to add:
A graduate student shouldn't need a fellowship in order to remain in the lab or in grad school, but a good adviser will let you apply for one or maybe even demand it. It gets your work organized, lets you see early on what it is like to write something in a given lab and if you get it, it will give you some security and also be a nice feather in your cap. Plus it is practice for that one you write for the post-doc.
If a professor recommends against applying for it, saying he has plenty of money, this might be the first of many times when he says,"I don't care too much about your career or CV, I want you in the lab, lab, lab generating data for me to put my name on."
If you are in lifescience, you can check get some info about grants here
http://crisp.cit.nih.gov/
In our biology department, there are two divisions. Developmental, Cell, and Molecular Biology works like what you've described--new grad students come in and shop for a lab and most students are supported by PI's grants.
In Ecology, Evolutionary, and Organismal Biology new grad students are admitted to school to join a particular lab based mostly on the opinion of that PI and most of us are supported by outside fellowships. In Just wanted to provide a different datapoint.
Hydro, there are programs here that also work like that- I generally tell people to avoid them if possible, since bad things can happen even easier. I have a collection of horror stories. Rotations, they are a thing of wonder.
Angrygrad, thanks for the link- I'll have to put it up.
Henry, we were encouraged to apply for NSF and NIH- I think HHMI wasn't offering that year- but absolutely, it is an excellent idea and a discouraging prof is a bad, bad sign. Good advice.
Commenting by HaloScan
