In Canada, most undergrad programs at University offer merit based scolarships (you do need above 9.0/10 GPA to even consider getting one though).
During my graduate program, I was actually funded by a provincial science funding agency.
At my University, in Canada, so few people applied for scholarships that often all you needed to do to get one was apply. Only applicant = award is yours!
Generally speaking, this is true of local targeted scolarships (e.g., scolarship for aboriginal students working in history). That is not the case for scolarships given by the federal and provincial funding agencies.
In US, does the NSF/NIH give out scolarships?
(Apologies for the double post, saw this after I wrote the other response)
Yes, they do. The NIH has 2-year Training Grants, which are typically a number of slots awarded to a University entity (a department, a graduate program, etc.). This covers tuition in some amount, and pays the student a stipend at the current NIH graduate student rate (it was $28,500/year when I started grad school, I think it's up to $29K or so now). They're awarded to students by the University entity that gets the slots, so the competitiveness for them varies a bit, and is generally very local in any case (but most programs get on the order of 2, maybe 4, slots a year).
The NSF has the Graduate Research Fellowship Program, which is a three year fellowship that awards $12000 to the University per year, and gives the student a $32,000/year stipend, which is increased every couple of years. The GRFP is generally considered a fairly competitive fellowship with about 2000 awards a year across the US.
There's also a handful of other fellowships through either private foundations (Hertz, some others), which tend to be even more competitive than the GRFP, and there's some other government ones, such as the NDSEG which is military related, and some field-specific Dept. of Energy fellowships.
To my knowledge, these are all for Ph.D. students, and don't apply if you're in a Master's program.
In general, the fellowships don't cover all of the 'official' tuition, and the difference between the tuition covered by the fellowship and the 'official' amount is covered by 'University Gift Aid'. This is, of course, just the University paying money to themselves, especially since the vast majority of credits taken by Ph.D. graduate students are "Research", aka doing your job for the University. Oddly enough, for most students who don't get fellowships, the 'gift aid' gets bigger... (and actually comes out of the advisor's grant money). The upshot of all of this is that in general, STEM Ph.D.s receive a small salary, rather than paying, for their degree.
(Source: I was fortunate enough to receive both a 2-year NIH training grant and a 3-year NSF Fellowship back-to-back.)