Is there an official list of objects in the sky? I would like to know how many known quasars there are in the Universe as well as x-ray binary black holes (not Neutron stars) but I don't seem to be able to.

I also want to know how many black holes have been found via gravitational wave detections but I think the answer to this question is 3.

Do you guys know any credible sources that list the objects in the sky via category and gives detailed info of how we found them (x-ray binary or gravitational waves or quasars).
Lists of astronomical objects are called "catalogues" and other answers have already mentioned the NGC catalogue of galaxies and nebulae. However, you asked about quasars and black hole x-ray binaries.

For quasars, there are several catalogues. None claim to be complete, as new discoveries happen all the time. The MILLIQUAS catalogue is a compendium of 460,222 type-I QSOs and AGN. It also contains about a million "Quasar candidates". (sources that may be quasars, 80% confidence)

For Black hole binaries, the difficulty is proving that the object is indeed a black hole. This usually requires calculating the mass of the accreting object, which is not always easy. The X-ray catalogues are classified by the mass of the normal stellar component. The High Mass x-ray binary catalogue notes objects that are "black hole candidates". It is in no way a definitive list, since no such list exists.