There are basicly 3 main tracts of Mesorah - sources of tradition (I would say in Breslov but let it be clear what Rabbainu said that everyone is going to be Breslov, thus this is Mesora of Judiasm) from Rabbi Nachman. The first step is Rabbi Nussun, although this was slightly disputed, nowadays this is basicly accepted by everyone. Rabbi Nussun himself did not become the Rebbe of Breslov. After Rabbi Nachman's passing Rabbi Nussun set out to visit the great Chasidic masters of the time to see if they could continue to guide him on the correct path of God. Rabbi Nussun surmised that none of the Masters were capable of properly guiding himself or anyone in the ways of God, whereas he, as little as he was in his own eyes, was equiped with the awesome teachings of Rabbi Nachman which have the power and capability of guiding everyone. Thus Rabbi Nussun of Breslov took upon himself to do his utmost to proliferate the ways of Rabbi Nachman.
There is some controversy as to whom is the strongest successor of Rabbi Nussun. There are two very strong contenders, Rabbi Moshe Breslover and Rabbi Nachman of Tulchin, and there is also a weaker route of tradition based on Rabbi Nachman of Tcherin - who frankly admitted to Rabbi Yisroel Karduner that he did not completely and properly master the tradition, and he therefore sent Rabbi Yisroel Karduner to service Rabbi Moshe Breslover. Rabbi Moshe was born due to a blessing of Rabbi Nussun, and as a young child he became an orphan r"l and he was raised personally by Rabbi Nussun. Rabbi Moshe, as a young child, was known to be a great prodigy. Rabbi Moshe was extremely modest and there is very little known about him!
Thus Rabbi Yisroel Karduner, one of the 36 hidden tzadikim - and in Breslov it was said that even had he lived in the time of Our Master Rabbi Nachman, he would have been considered a novelty - was the succeeder of the mesora from R' Moshe Breslover, and he in turn gave over the tradition to his protege, Rabbi Yisroel Dov Odesser also know as the Sabba. The Sabba said that until the coming of Mushiach there will not be any known Rabbis that people could rely upon.
Following the other route of the tradition through Rabbi Nachman of Tulchin, there is a short gap, because his greatest successor seems to be his son Rabbi Avrohom Ben Rav Nachman, and Rabbi Avrohom writes clearly that he did not merit to fully study the tradition from his father, writing that until 3 years after his father's passing he regarded his father as a somewhat simple Jew (in fact his father was 20 years old when he met Rabbi Nussun, and Rabbi Nussun warned him not to learn the talmud, just the Chumash and Prophets, halacha, agada, kabala, and ethics, lest he would become a 'byzenne lamdan' - austere scholar. Obviously later in life he merited to have great mastery of the Talmud) albeit with tremendous mesiras nefesh - self sacrifice.
Rabbi Avrohom writes that 3 years after his father's passing he realized that the world doesn't operate in a vacum, and if his father had such great mesiras nefesh he obviously was also 'hooked up' and was very qualified to be the strongest successor of Rabbi Nussun. Rabbi Avrohom then gives some arguements trying to prove that his father was even a stronger successor that Rabbi Moshe Breslover. Being that the Saba said about Rabbi Avrohom that he was an איש אמת a man of inegrity, it is not so much for us to debate his arguements (which definitely have plenty room to be disputed), especially since it doesn't so much concern us. There is no strong succession from Rabbi Nachman of Tulchin on, because R' Avrohom himself says that during his father's lifetime he did not relate and service his father as the master of the tradition, just a simple committed Jew. Also the Sabba said that the amazing relationship he, the Saba, had with R' Yisroel Karduner could not have been had even with R' Avrohom. Thus the strongest mesora from Rabbi Nussun is clearly traceable upwards, from the Saba, Rabbi Yisroel Dov Odesser, up through Rabbi Yisroel Karduner, up through Rabbi Moshe Breslover.
There is a tradition based upon Rabbi Avrohom Ben Rav Nachman of Tulchin. R' Avrohom impacted many Breslovers including Rabbi Bender (who was a member of one of the lower groups in Uman mentioned in Yimay Shmuel), most of the so called Breslover Rabbi's today can be traced to Rabbi Bender. Rabbi Bender started kicking people out of Breslov when they showed desire to be Rabbis or when they attracted 'crazy' people to Breslov.
The weakest path of tradition today can be traced through R' Shternhaltzer, a grandson of R' Nussun, the mesora of the original American Breslovers, such as Rabbi Rosenfeld and his student Arye Kaplan, come from him. This Mesora is obviously the weakest tracing themselve through Rabbi Nachman of Tcherin. Its followers made many concessions to society that are blatantly against Rabbi Nachmans way, e.g. they cut their beards!