You've been there. We all have. Maybe you've learned how to curb your errors, but if you're reading this, you probably haven't cut them down as much as you'd like.
Two keys to minimizing stupid mistakes are developing good habits and organizing your work.
You’re certainly familiar with the biggest sources of error: misreading the question. The surest way to beat that is to read the question more than once. If you get stuck on the problem, read it again to make sure you have it right. Most importantly, once you have finished the problem, read the question again, paying close attention to exactly what the question asks for. Only after you've verified that you've found exactly what the question seeks should you write your answer down.
This repeated question reading may seem like it will take a lot of time, but if you make it a habit – even do it on the easy problems - you'll get to where you do it so fast you won't even notice it.
Read problem - do question - check that your solution is exactly what the question asks for - mark your answer - next question.
Next habit to develop (and this one's even harder for most people): write clearly. If your work is chicken scratch, you’re doomed to misreading numbers, messing up signs, and so on. Notice in the diagram at the right that it doesn't take much to make your writing a little neater. With only a little practice, you'll soon be just as fast with more legible writing (even faster, in fact, since you won’t have to think 'what does this say' when looking at your own work).
Organizing your work will go a long way towards stomping out stupid mistakes. Right now, your scratch paper after a tough test probably looks like the paper on the left. Scrawl everywhere - each time you started a new problem, you just looked for an open space on your scratch paper and started writing. Instead, imagine your paper looked like the paper on the right below. Work for each question is segregated in its own numbered box (or on its own sheet of paper for harder tests). Your work is easy to find when you want to check your solutions. But the biggest benefit is that forcing organization on yourself will make you subconsciously more organized in other areas (of your test taking - this won't help a bit with the mess in your room). You'll make fewer errors copying equations over from the test, and fewer mistakes recording your answers.
You might ask, "Won't I waste tons of time making boxes for my work?" The answer is no: prepare your scratch paper beforehand. Make your grids & number the boxes before the test starts.
Good organization makes the last good habit easier. Be aware of your time. With 5 or so minutes left, review the questions you've already answered. Cruise over your scratch work and make sure what you have there matches what you wrote on the answer sheet. The cells on your scratch paper should be numbered and have the answer circled, so doing this check should be easy and fast. After this pass, you can go back to scrambling after those last few problems you haven't solved yet.
You may be thinking, "I'm a messy person. I can't do all that." I've been there. I'm still a generally messy person - clothes on the floor, books and papers scattered around my desk. But after many near misses in various competitions, I changed my test-taking strategy. Other people thought I'd suddenly gotten smarter or that I studied harder. Maybe I did, but mostly I just got organized.