As the title asks, how many bilateral covenants occur in scripture between God and His creation (this does not include bilateral covenants between humans and humans, e.g., a marriage covenant between a man and a woman, or the covenant between Avraham and Avimelekh).
Although we often find various covenants spoken of in scripture such as the covenant with Adam, Noah, Abraham, Moses, etc. there are really only two covenants between God and man. The remaining covenants are merely modifications, progressions, or what might be called dispensations of two single covenants.
The first covenant is the covenant of works established in the creation of Adam and Eve. By virtue of being created mankind was under natural duty to love and be absolutely obedient to God. If Adam were to fully obey God, during some period of testing, he would have obtained eternal life for humanity. That is to say that Adam was a federal head and representative of the covenant of works with respect to humanity. Upon Adam's obedience or sin mankind would have been imputed with death and sin, or righteousness and life forever.
Upon the breaking of that first holy covenant, resulting in the death and moral fall of all mankind a new covenant was required according to the love and mercy of God as well as his justice in being unable to forgive sin. The second and final covenant was the covenant of grace being promised through the seed of the woman, who was to crush the head of the serpent and restoring the creation to God.
These two single covenants run parallel throughout the redemptive history of scripture and never absolutely join as they are opposed to one another, in that one is of works and one is of grace. From Adam to the end of the world, man has and always will have the choice of remaining condemned under the covenant of works, or through faith in the promised seed, obtain new birth and eternal life under the covenant of grace. Before Christ men believed in the new covenant as a promise, after Christ men believed as a historical work.
That there was a long and steady degeneracy of the original knowledge of the covenant of works is evident in the progression of idolatry. That there was a slow developed progression of the knowledge of the covenant of grace until the climatic revelation of Jesus Christ and the full clarity of the gospel and law as proclaimed in the Epistles of the New Testament, is evident in the history of Israel and the gospel extended to the gentile world. To properly understand these two covenants and to properly distinguish them from one another is the entire work of scripture and theological study resulting in the purest glorification of the Trinity.
Under the Old Testament, the nature of law degenerated greatly in the world until the time of the flood. In the calling of Abraham, God called a people to preserve the understanding of law. In parallel, God also called out a people to develop the knowledge of the promised Messiah to relieve the death that the law proclaimed. This temporary measure, of a church based on race, was to safeguard religious knowledge in a wicked fallen world. The world had little light left to understand the original covenant of works and the promised new covenant. This preservation also preserved the lineage of the great Messiah to come to that world according to the promise and various prophecies in this regard.
By re-starting humanity, in a sense, under Noah, purging the filth of the world, God modified the covenant of works and re-established the covenant of grace. By killing the whole world law was realized and a renewed promise in the rainbow was raised high up into view. Through Abraham, both covenants were again further strengthened in separating a people unto holiness, condemning the world through law, while also promising the seed more explicitly to bless all nations.
In calling Moses to establish an outward form of religion first re-enforced the covenant of works, in its condemnation of sin. Also the ceremonies and theocracy figuratively further established both the law and the covenant of grace. Both covenants were made stronger in men's understanding, while yet being of opposed means, one of works the other of faith. Having established both parallel covenants under one system of practice, the covenant of works under the Law and the covenant of grace to redeem from that Law, a greater light of these two covenants was provided. Furthermore the Law it its condemnation and death, was no through figures and ceremony directed and officially pointed unto a new end and new covenant. The new covenant was to be the end of the old covenant and so the theocracy, ceremony and outwardly written laws under them, a temporary period until the full revelation and ratification of the new covenant through the shed blood of Messiah. By this means a full and harmonious revelation was established becoming the absolute rule and guide for God's called out people according to Jewish race. This people was established not in service to themselves or their own greatness but to the world at large, for who both covenants were designed.
In the fullness of time, that is when the world and the Jewish church had both fully fallen into an absolute failure to find any means of saving themselves, Christ appeared. In his death, he destroyed the demands of the first covenant, ended the temporary figurative and theocratic elements of the dispensation of a physical church, provided the full accomplishment of the priestly works to ratify the new covenant, rose and entered into glory to act as the new federal head and representative of a new humanity. By raising up as a King, Priest and Prophet, sending his Spirit into the world to continue his work until the end of the age, the first covenant was officially destroyed and the new covenant absolutely ratified before the court of the Father and to the delight of angels and men. The Spirit now works upon the consciences of men, saving those that perish by proclaiming this gospel with certainty to the eternal salvation of any soul that will but believe.
Conclusion: There are only two covenants from Adam to the end of the world, but they have undergone modifications and progressions in the destruction of the first broken covenant and the development of the second perfect covenant, according to the incomprehensible wisdom, power and love of God.
Note: I should probably add scripture references to this post but I just typed it quickly as it came to mind. As each sentence can be referred to by numerous biblical references I am afraid I would have never attempted to answer as it would have taken me hours. I would not usually do so but I am relying on the reader to have some knowledge of the bible to know that each step of my argument can refer to very numerous biblical references. I have done this as the question really requires a broad brush of scripture as it is a very difficult and broad question.