Spoilers up to and including Half-Blood Prince.
This might seem a bit out there, but I've become convinced that the Full Body-Bind (Petrificus Totalus) is plain and simply better than Stupefy. Let me establish their properties:
The Full Body-Bind is:
- Seemingly pretty easy to learn. Hermione famously used it on Neville in her first year (Philosopher's Stone, chapter 16) and we see it used many times by members of the D.A. in the Department of Mysteries. I cannot recall any point in the books where anyone mentions struggling to learn it.
- Very subtle. Harry cast it twice in chapter 28 of Half-Blood Prince while chasing some Death Eaters that aren't far away, but there's not any reference to them noticing him until much later. Furthermore, when Dumbledore cast in on Harry in chapter 27 (the spell used is named in chapter 28), he literally didn't know what hit him, and Draco made no sign of hearing a noise when he entered the room at the same time as the spell being non-verbally cast. Aside from a reference to someone failing to block it - implying that there's something to actually block - I cannot recall any reference in the books to this spell creating any sound or sight that would suggest that it's been cast. In fact, although it can miss (e.g. it happens in chapter 31 of Deathly Hallows), I cannot recall it ever being dodged.
- Able to hit multiple targets? This confuses me, but I can't find any other way to read this quote from chapter 26 of Half-Blood Prince: “Petrificus Totalus!” Harry bellowed again, backing away as he swiped his wand through the air; six or seven of them crumpled, but more were coming toward him.
- Here's the most important part: Aside from one notable case where the death of the caster ended the spell, it appears that the Full Body-Bind lasts until it is counter-cursed (e.g. Half-Blood Prince, chapter 28: "the full Body-Bind Curse Dumbledore had placed upon him lifted, known that it could have happened only because its caster was dead"). This is fantastic on several grounds: 1) If the opponent's allies don't know the counter-curse, there's every chance that you've just killed the target. 2) Casting the counter-curse makes your opponent's allies open to being hit by you. 3) The target is completely out of the fight until the counter-curse is used. Unlike Stupefy, they're not going to unexpectedly get back up.
- Useless with non-humanoids? We know that Stupefy works on non-humans, e.g. Goblet of Fire shows it working on a dragon, but given the position that the Full-Body Bind puts its target in, the comments have pointed out that it might be safe to guess that it won't work properly on non-humanoids.
And Stupefy is:
- Tough to learn. Chapter 29 of Goblet of Fire tells us that Harry had never used it until he was practicing for the final Tri-Wziard task and chapter 21 of Order of the Phoenix shows a sixth-year, Cho Chang, telling Harry that she's never managed to stun anything before.
- Absolutely blatant. As chapter 31 of Order of the Phoenix shows us, it produces a beam of red light that can be seen from a tower away and has been known to cause the target to be illuminated and lifted high off of their feet. This lack of subtly also makes it dodgeable, as shown time and time again in the battle in the Department of Mysteries.
- Only able to hit one target at a time. I know of no example showing otherwise.
- Of unreliable, and sometimes short, duration. This is again the most important part. In chapter 35 of Order of the Phoenix, Lucius Malfoy is seemingly hit by it, but he recovers merely two pages later. When Professor Flitwick is hit by it in Half-Blood Prince, chapter 29 shows that it took him seemingly a few hours to recover.
- Optionally counter-cursable. Chapter 29 of Goblet of Fire again: Ron was "Stunned and reawoken by Harry for the fifth time in a row" while Harry was practicing the spell.
- A knock-out. The comments and the answers have rightly pointed out that this is sometimes better than keeping your target conscious, e.g. if you don't want them to hear what you're up to.
- Stackable. This is the one thing that I can definitely give to Stupefy over the Full-Body Bind, which as far as we know isn't stackable (what would stacking it even mean?). Goblet of Fire makes it clear that hitting a dragon with many simultaneous casting of Stupefy will take it down, and Professor McGonagall was hospitalized when she was hit with four beams at once in Order of the Phoenix.
So, aside from the fact that multiple castings of Stupefy on the same subject seem to increase its effectiveness to near lethal levels, why is it used so much more frequently in the series than the Full Body-Bind, which seems to be superior in many other significant aspects?