Below is a list of major trinitarian heresies, with definitions, and links to relevant articles:
Arianism: The heresy of Arianism defines the Son of God as a creature and denies His eternality. Arians believed that God created the Son in time as the first and greatest of His creatures, through Whom He then created all else. Arianism affirms many aspects of orthodox trinitarianism, including the pre-existence of the Son, the Monarchy of the Father, and the existence of three really distinct persons of Father, Son, and Holy Spirit. See Does teaching the Father is the one God undermine the divinity of Christ?, Eternal Generation Proved from the Scriptures, Begotten Vs Created, and Arianism Refuted.
Modalism: The heresy of modalism arose in the late second century, and was at first popularized by the teaching of Noetus, Praxeas, and Sabellius. Some early Roman Popes also taught modalism. Modalism teaches that in order for there to be one God, there can only be one divine person; therefore the Father, Son, and Holy Spirit are all one and the same person and reality. Modalists deny that the Father, Son, and Holy Spirit are really three distinct persons, realities, or individuals. Modalists instead teach that “Father”, “Son”, and “Spirit” simply refer to different modes of manifestation of one divine person, and are thus not three persons, but three “personas”. Some modalists teach that the “Father”, “Son”, and “Holy Spirit” are three internal modes of subsistence of the one divine person, making the modes internal and natural to the single individual. By making the Father, Son, and Holy Spirit out to be a single individual or reality, modalism teaches that the Father became incarnate and died on the cross, since the Father is ultimately one and the same reality as the Son- for this reason, modalism has also been called “patripassionism”. Modalism affirms aspects of orthodox trinitarianism, including monotheism, and that the persons of the Trinity share one and the same divine nature. See Why Are We Monotheists?, Modalism, Tritheism, and Subordinationism; Your Only Three Real Options Regarding the Trinity, Myriad Modalists, and Does the Name ‘LORD’ Being Applied to Multiple Persons of the Trinity Mean They Are All Really One Person?.
Semi-Modalism: The heresy of semi-modalism arose in the early fifth century as the church’s theology was morphed in over-reaction to Arianism. Augustine of Hippo was one of the major early proponents of this heresy. Semi-modalism, like modalism, teaches that there is one God because there is only one divine person; thus the Father, Son, and Holy Spirit are all one and the same person, individual, or reality. Unlike modalism, semi-modalism also affirms, in contradiction to the previous point, that the Father, Son, and Holy Spirit are really three distinct persons and individuals, not merely modes of manifestation or subsistence of one person. Semi-modalism thus exists usually as in inconsistency in belief, and not as a consciously-held theological position. Semi-modalism has been frequent since the fifth century as many trinitarians who affirm the real existence of three distinct persons fell into simultaneously thinking of these three persons as ultimately being one person, the one God, sometimes called “God the Trinity” or “the triune God”. Semi-modalism holds much in common with orthodox trinitarianism, affirming monotheism, the real existence of three distinct persons of Father, Son, and Holy Spirit, and that these three persons all share one and the same divine nature. See Modalism has evolved, Semi-modalism as the Greatest Problem Facing the Church Today, Semi-modalism and the Introduction of a Four-Person Trinity, Van Til’s Views on the Trinity, Augustine’s Trinitarian Heresy, The Grievous Error of the Fourth Lateran Council, Semi-modalism’s Absurdity in Light of the Simplicity of the Divine Nature, and Is It Okay to Call God “She”?.
Tritheism: The heresy of tritheism has not been common through church history. Tritheism teaches that there are three Gods, and denies monotheism. Tritheism treats the three persons of Father, Son, and Holy Spirit as three Gods, by teaching that the persons are separate from one another, and equal and identical to one another ontologically. In doing so, tritheism teaches that there are three unbegotten persons, three supreme uncaused Causes of all, and three supreme Authorities over all, thus making there be three Gods. Tritheism holds some points of doctrine in common with orthodox trinitarianism including that there are three distinct divine individuals or realities, and often that all three persons share the same divine nature. See Modalism, Tritheism, and Subordinationism; Your Only Three Real Options Regarding the Trinity, Does Rejecting Semi-Modalism Lead to Tritheism?, Why Don’t the Son and Holy Spirit Constitute Second and Third Gods?, Why There is Only One God: An Introduction, Why There is Only One God: One Supreme Cause, Why There is Only One God: Headship, Why There is Only One God: One Divine Nature, and Why There is Only One God: Relational Unity.