Personal injury attorneys routinely work on contingency. Look for the ad that says they don't get paid until you recover. If they are telling you that they will not take your case unless you pay them up front, that suggests that they don't think you have a very good case, or alternatively that your matter is simply not suitable for a contingency recovery. For example, attorneys are strictly prohibited from charging a contingency fee in dissolution cases.
Nearly all personal injury attorneys work on a contingent fee basis for a variety of reasons. If you are having trouble finding an attorney to take your case it could be that the liability, damages or some other aspect of the case is not strong enough to merit the attorney's taking the case. Remember, that under the contingent fee arrangement, the attorney's office does all the work without any pay whatsoever unless or until the case either settles or goes to trial and a positive verdict. For that reason, attorneys who work on a contingent fee basis must screen out cases that they do not believe warrant taking on the risk.
I truly wish you the best.
This answer is provided for informational purposes only. Actual legal advice can only be provided in an office consultation by an attorney licensed in your jurisdiction, with experience in the area of law in which your concern lies.
Attorneys work under several different payment schedules. Contingency fee agreements are usually undertaken when someone has been injured, and as others have said, are prohibited in certain other types of matters. If you have been injured, most contingency fee attorneys are going to be listed on the internet or the yellow pages, etc. under "plaintiff's" or "personal injury" or "accident" attorney. Some attorneys work throughout the state and do not charge for an initial consultation. For ease, you might also want to call someone who has a informational website and a toll free number, which might help you explore your options and increase your chances of being able to get the help you need. Ask family and friends also because personal experience is highly valuable, and you must trust your attorney implicitly and know that your interests are being protected. Sometimes a trusted advisor in one area of law will be able to refer you to an able colleague. Good luck in your search.