What is the Scriptural basis for Modalism?

The umbrella term for the doctrines that include Modalism is Unitarianism.

The scriptural support for Modalism is the same as that for the Trinity, except they interpret the scripture as speaking of God as one person manifest in our dimensions in differing modes. I would argue that the Unitarian interpretation involves ignoring multiple key scriptures that evince three separate persons, such as at Jesus' baptism where the Holy Spirit comes down and the Father speaks:

9 In those days Jesus came from Nazareth of Galilee and was baptized by John in the Jordan. 10 And when he came up out of the water, immediately he saw the heavens being torn open and the Spirit descending on him like a dove. 11 And a voice came from heaven, “You are my beloved Son; with you I am well pleased.”

In Unitarian thinking, therefore there is one person who, in the OT, is revealed as the Father God, and in the New Testament is revealed first as the incarnate Christ, and then, after the ascension, as the Holy Spirit.

They contest the orthodox claim that there are three persons in the Godhead, and claim instead there is one person who, at different times, wears a different "mask", if you will.

Closely related is Patripassionism, which takes Modalism further and says that the Father suffered at the cross (and, presumably so did the Holy Spirit) since each is merely the same person in different guise.

The Trinitarian doctrine, is that God is one in nature and three in persons. Conversely, the Unitarian doctrine is that God is one in nature and in person.

The United Pentecostal Church International, who are a denomination of Oneness Pentecostalism, and adopt the doctrine of Oneness, which is essentially the same as Modalistic Monarchianism, offer the following in support of their Doctrinal Foundation:

The Oneness of God

God is absolutely and indivisibly one (Deuteronomy 6:4; Galatians 3:20). In Jesus dwells all the fullness of the Godhead bodily (Colossians 2:9). He is the self-revelation of the one God, the incarnation of the full, undivided Godhead (John 20:28; I Timothy 3:16).

God has revealed Himself as Father (in parental relationship to humanity), in the Son (in human flesh), and as the Holy Spirit (in spiritual action). (See Deuteronomy 32:6 and Isaiah 63:16; Luke 1:35 and Galatians 4:4; Genesis 1:2 and Acts 1:8.) The one God existed as Father, Word, and Spirit before His incarnation as Jesus Christ, the Son of God; and while Jesus walked on earth as God Himself incarnate, the Spirit of God continued to be omnipresent. However, the Bible does not teach that there are three distinct centers of consciousness in the Godhead or that Jesus is one of three divine persons.

Jesus is true God and true man as one divine-human person. We can distinguish these two aspects of Christ’s identity, but we cannot separate them. The Incarnation joined the fullness of deity to complete humanity.

Jesus possessed all elements of authentic humanity as originally created by God, without sin. Thus we can speak of Jesus as human in body, soul, spirit, mind, and will. (See Matthew 26:38; Luke 2:40; 22:42; 23:46; Philippians 2:5.) According to the flesh, Jesus was the biological descendant of Adam and Eve, Abraham, David, and Mary. (See Genesis 3:15; Romans 1:3; Galatians 3:16; Hebrews 2:14-17; 5:7-8.) We should not speak of two spirits in Jesus, however, but of one Spirit in which deity and humanity are joined.

Christ’s humanity means that everything we humans can say of ourselves, we can say of Jesus in His earthly life, except for sin. In every way that we relate to God, Jesus related to God, except that He did not need to repent or be born again. Thus, when Jesus prayed, submitted His will to the Father, and spoke about God, He simply acted in accordance with His genuine humanity.

As Jehovah manifested in the flesh, Jesus is the only Savior (Isaiah 45:21-23; Matthew 1:21-23). Thus, Jesus is the only name given for our salvation (Acts 4:12). The Father was revealed to the world in the name of Jesus, the Son was given the name of Jesus at birth, and the Holy Spirit comes to believers in the name of Jesus. (See Matthew 1:21; John 5:43; 14:26; 17:6.) Thus, the apostles correctly fulfilled Christ’s command in Matthew 28:19 to baptize “in the name [singular] of the Father, and of the Son, and of the Holy Ghost” by baptizing all converts with the invocation of the name of Jesus.

Source: Our Doctrinal Foundation; United Pentecostal Church International