I would like to apply for a job which was posted online. After research, I found out that the exact same job is posted by different recruiting agencies (at least 12 oe more). Do I look for the most reliable one and apply only once (considering that they might not be the best choice and I will be sorted out), or do I send out my resume to all of them (considering that the company will sort me out because they have the same application 12 times). I was not able to find the employer.
Do NOT apply through multiple recruiters. Agreements with recruiters often include that they are the only one allowed to represent you to a given company (meaning you can't apply directly either). Going through multiple recruiters for the same company is viewed as a breech of ethics by many people. If a company gets your resume through multiple sources, they are much more likely to simply throw you out as untrustworthy or not worth the risk of a lawsuit with the competing recruiters.
You might find this question interesting as an example of what can happen when you go through multiple recruiters.
I would not apply to all of them. In my opinion, the company would get extremely suspicious if they saw your resume applying for the same job 12 times. If they were interested in you to start with, seeing this behavior may cause them to reconsider asking you for an interview. Furthermore, some recruitment agencies will ask that you only work through them--whether or not you're willing to comply with this is up to you, of course.
At best, you still get an interview and nobody notices/cares. At worst, they eliminate you as a candidate entirely, possibly burning bridges in the process. The risk isn't worth the gain, in my opinion. If you're qualified, one application should be sufficient.