Many of the people and organizations that I work with are familiar with my problem solving technique, and 'Problem Solved!' exclamations, and there are some important details to cover about linear though and planning that are often less visible. So this will be a step-by-step problem solving guide.
1) 'One problem at a time' - Identify and categorize the problems.
Yes, the online time management gurus are currently in favour of unitasking, and that doesn't really mean do only one thing - it means take care of one problem at a time. In the flat tire example this means before you run to a mechanic you will need to make a temporary fix to your vehicle. To make a temporary fix you will need the equipment. Hope you planned ahead. The rundown:
Equip your vehicle (done); Tire is flat (yes); Bring spare tire, jack, and tire iron to flat (you are here); Jack vehicle; loosen nuts; Etc. (Actually you may want to see if you are supposed to lightly loosen some of the nuts before lifting the vehicle, read the manual)
2) 'One Man, One Mission' - Identify the leader and potential solutions.
Only one solution will work, the others could have but only one can be implemented. Without a clearly defined leader there may be disagreement, which reduces the number of solutions coming in. Sometimes the leader is not the expert, they are often the person that you'd ask 'is this a good idea'. Their response defines your course of action.
More solutions are welcome until the group or leader is certain. Dissent may still appear, but unless they can prove that the current solution isn't going to work they really just need to help or get out of the way.
3) 'The Fail Train' - Problems arising from last solution.
Remember that person in your group that said 'that's not going to work', sometimes they're right and it's too late for gloating because you have to start problem solving from a new point of view. Typically you get three strikes, if the solution creates more problems by the third round it is time to stop and seek out an expert. If you think you are the expert then its time to get a grant and start your research. Don't forget to publish your results 'expert'.
Stop the fail train through reduction. If the car doesn't work then find a way to live without it, or get rid of the old one that is unfixable. So many people are looking for the additive solution (eat this to stop being fat), when the solution is to reduce (put down the 'metabodonut' it's not working).
4) 'Problem Solved!' - It is important to track progress and document results.
Documentation helps to make sure this problem doesn't happen again, usually you're repeating someone elses fix. You do not need to post your tire changing guide, but you do need to run down your checklist. Your checklist is where the paperwork starts. That list will get longer, and your successes will grow in number.
I don't know about all of the problems, only the ones that people bring to me. Your problems are going to be different. If you want to be happier though it is a good idea to take things step by step, one problem at a time, lead your mission, and avoid the fail train. Then you will be happier and the world will be a better place because of that.