The answer to your question lies in the answers submitted here up to this date. Those who have answered this question positively are those who know that they have weaknesses themselves and are honest and secure enough to expose vulnerability to you so that you might learn something. They are also smart enough to know that whatever weaknesses exposed here aren't going to be exploited by simply posting an answer.
Others here have given you negative advice and are exposing their weakness unwittingly. The weakness the negative answer exposes is that they desire more than anyone, to use people's weaknesses against them. The evidence is obvious. There's a post that drew their interest. Why would a post that is offensive to them draw interest unless THEY were searching for the very same answer to the question you are asking?
In the answers here the weaknesses abound. There are those who must type with the "cool" style so that they "fit in" with their clique. Fitting in is very important to some. You may conclude that finding what a person considers to be "important" is a weakness in itself in addition the specific "important" subject. Almost everyone considers something to be important such as the several posts here that fear you might exploit someone's weakness. These folks fear exploitation. You might conclude that fear is a weakness.in addition to the specific fear that they have.
Now, you have 2 very general leads as to what people's weaknesses are. Further study will require that you make a list of all known possible weaknesses in people. Then, you can decide what weaknesses certain people exhibit with just a little study. Work from the general topic and then try to fill in some specific categories beneath them. With an outline like that you can learn to spot in little time what weaknesses you are dealing with.
A Psychology class will help in learning what weaknesses are and how some weaknesses can actually be strengths.
A note to the negative posts: Folks, people want to learn things. Especially things that seem to be hidden from them. It's simple human nature. Allow other people to learn even if you don't openly approve. Everything has a value even if you, personally might not be able to see it in that moment. The other point is that it isn't healthy to assume the person asking the question here is asking it for nefarious reasons. The reason wasn't given. Often people will feel more at ease asking questions in a public forum because all other avenues have been exhausted.
For instance, a "weakness" could be someone's strong desire for chocolate. That's one type. Or a weakness could be an easily hurt-able body part; such as a bum knee. Again, it might also be a feeling inside someone that can be hurt easily by another. Like....If you've been hurt before by someone telling you you have an ugly nose. Then the next time someone says that to you, it hurts more, because it's your weakness. It makes you extraordinarily sad. Also, if someone's afraid of bugs, and they see a bug on their wall that person may back off or panic. Then you would know that is their weakness. Even roses can be a weakness. If someone really loves them & you give them roses, they may swoon when they receive them. This too can be regarded as a weakness. In ending, and in answer to your question...I'll say that you have to watch closely. Asking someone doesn't usually get you to a correct answer. Nobody wants to reveal their weaknesses;unless they are the "good" ones. Rarely, you could fool someone into telling you what theirs is...but why would you want to know ?
It is also the case that, after a major life change, gain, or loss, sometimes people's weakness changes; sometimes too they lose weaknesses.
Ultimately there are probably more useful pieces of information and more effective tools, given you are seeking a general answer to this question in such a public place.
WARNING: What you are about to read might come off as twisted and disoriented.
To mentally and subconsciously destroy someone you must think like them. Use what you have against them. Any information, any strengths. Weakness could be mental or physical. Maybe even verbal. See, this depends on how much you hate the person you are trying to overcome.
Mental: People are influenced on what happens in their childhood, huge stages that impacted their life. Break down those stages and turn it against them like a tidal wave. Don't hold back, that is a sign of weakness on yourself. First, level the playing field. Blackmail is always a very good way of taking someone down. Burning them. If they have any problems in their lives, exploit them. Use the people around you to melt them into nothing by a crispy consciousness that could be beaten down into what you want it to.
Physical: This part is easy. You need to analyze what this person hates. And I mean, really hates. I mean, "Can't withstand it" hate. Me, for instance. I can't take the sound of Styrofoam squeaking against itself, or the sharp deathly noise of a wet shoe skidding on the ground. Or the corniness of those old TV shows where the audience would laugh at anything they here. Or clap. Disgusting.
How to protect yourself: Do not show any sign of weakness. Do not mess up, talk slower if you have to. If your really in a pickle, fake a phone call to buy time. Always be the smartest guy in the room. And don't be cocky. Don't make assumptions on others on a high level. Like how if a person comes in that you are attracted to, ignore those feelings. Find out if the character is trustworthy, what their weakness is. Be on top of everything, be ten steps ahead. Like in chess. Life is a chessboard. You are the king. But the only thing different between life and chess is that the king controls all of the pieces. If he's wise enough. Studying history would also help, seem smart when you really aren't. Over-exaggerate when you need to. When you look upon people, make subtle assumptions. Ones that are could have a chance of being 90% true. Beautiful, attractive? Possibly rich, uses people to their advantage. Or just an idiot born that way. Glasses? You can sabotage them and create a weakness. Ugly? Probably intelligent, knows what their doing when push comes to shove. Got a ring on their ring finger? Married, obviously. And so on and so forth. Just be prepared for the worst. Always be 10 steps ahead in war.
Ok...in the military it is the job to find the enemies' weakness....but why on earth would you even consider this in your life. Has someone done something so awful to warrant this?