What churches identify themselves as being Modalist?

Aside from Trinitarians, people who hold other Christological views tend to explain their views rather than just label them with a term, so you won't find a denomination saying on its website "We're Arian" or "We're Modalist" or "We're Samosatosians" or "We're Patrispassians." Its not going to happen, because how do these groups come to these conclusions? From Scripture, not from following any man, so they think of their view as merely being the scriptural view.

Modalism is saying "God is one person; but he's called the Father sometimes, called the Son sometimes, called the Spirit sometimes," and further they will usually also say "God is one person: Jesus Christ." So if anyone says that (i.e. "God is one person: Jesus Christ") then they are a modalist for sure.

But some Oneness may not be modalists, as some of the Sabellians and other monarchians such as Ebionites, Marcellians, and followers of Paul of Samosata, etc. also weren't modalists but rather were patripassians. What they believed was that God is only the Father, the Spirit is a vehicle or energy by which the Father dwells among men, and Jesus is a man in whom God dwelt (from the baptism forward) in all his fulness but as something above and beyond his full manhood (i.e. he didn't replace his soul or intellectual faculty, so that Jesus was not God but just a man with God tagging alone inside him). Because these say Jesus was not God, but only a man with God dwelling in him (but not controlling his body, Jesus' human soul still having full control) and since the spirit is hardly viewed as a person but rather as some sort of apparatus for God's dwelling in Jesus, this is certainly not modalism. Yet it is monarchianism, and I've seen at least one Oneness Pentecostal hold to a view like this. [Some may have also seen God as taking over Jesus' will. There's a bit of a range in belief that's possible with this stuff.]

Of course, all of these were labelled as heretical by the Catholic church...but I don't see why Protestants who do not derive directly from the "magisterial reformation" should even care about that. The magisterial side of the Reformation (Luther, Calvin, Zwingli) continued some Catholic traditions like infant baptism and continued to give the Catholic councils some doctrinal authority. But why should Baptists, Pentecostals, etc. who derive more from the Anabaptist side of the Reformation even pretend to care what any Catholic council had to say on anything? To do so would be pure hypocrisy.

Trying to find a good definition of modalism almost seems to be a moving target. Not unlike a lot of other definitions. The UPCI as a whole does believe that God reveals himself in different manifestations, relationships and/or operations, BUT, not one minister in the UPCI that I know of believes that as God operates in one capacity He necessarily ceases to be something else that He has been. For example, when God was manifest in the flesh, He most certainly did not stop being the Father. God is always and at all times a spirit and does not cease to be that. As a practical illustration, I am a father and just because I talk to my parents as their son does not mean I stop being the father of my son or cease to be a man.

The label modalist has been so strongly stigmatised over the millennia that we shouldn't expect to find anyone who calls themselves one now. Regardless of whether it is in reality a correct or incorrect understand of God, in our language it is basically defined as a heresy. And no one wants to be known as a heretic.

Sometimes people have reclaimed stigmatised labels (such as the LGBT community reclaiming queer), but those who believe in modalism have instead come up with new synonymous terms, one of which is the Oneness of God. A church which proclaimed that they taught modalism would scare away any potential visitors, but a church which proclaims the Oneness of God? That sounds more enticing.

David K. Bernard is the general superintendent of the United Pentecostal Church International. In his book The Oneness of God he plainly and openly teaches modalism, and declares that Oneness is a new synonym for it (emphasis added):

We also explore the relationships and distinctions among the three terms of Father, Son and Holy Ghost. Do these terms identify three different persons or personalities in the Godhead? Or do they indicate three different roles, modes, functions, or offices through which the one God operates and reveals Himself? (p125)

In a similar way, the Holy Ghost or Holy Spirit is not a separate person from the Father any more than a man and his spirit are separate persons. Holy Spirit just describes what God is. First John 5:7 says that three bear record in heaven; that is, God has recorded Himself in three modes of activity or has revealed Himself in three ways. He has at least three heavenly roles: Father, Word (not Son), and Holy Ghost. Furthermore, these three roles describe one God: “these three are one.” (p141)

The Bible speaks of Father, Son, and Holy Ghost as different manifestations, roles, modes, titles, attributes, relationships to man, or functions of the one God, but it does not refer to Father, Son, and Holy Ghost as three persons, personalities, wills, minds, or Gods. (p144)

The modalist doctrine is usually explained simply as the belief that the Father, Son, and Holy Ghost are only manifestations, or modes, of the one God (the monarchia), and not three distinct persons (hypostases). (p248)

Basically, modalism is the same as the modern doctrine of Oneness. (p318)

Thus Oneness is a modern term basically equivalent to modalism or modalistic monarchianism. (p321)

Sabellianism. ... The doctrine is basically equivalent to modern Oneness. (p324)

The UPCI website says that one of their doctrinal distinctives is the Oneness of God. It also has a tract arguing for the doctrine. The words 'modalism' and 'modes' aren't used, but clear synonyms are (emphasis added):

1. Is the word trinity in the Bible? No.

2. Does the Bible say that there are three persons in the Godhead? No.

3. Does the Bible speak of the Father, Son, and Holy Ghost? Yes.

4. Do these titles as used in Matthew 28:19 mean that there are three separate and distinct persons in the Godhead? No, they refer to three offices, roles, or relationship to humanity.

13. Who is the Father? The Father is the one God, particularly as revealed in parental relationship to humanity. Deuteronomy 32:6; Malachi 2:10.

14. Where was God the Father while Jesus was on earth? The Father was in Christ. John 14:10; II Corinthians 5:19. He was also in heaven, for God is omnipresent.

46. If God and the Holy Ghost are two separate persons, which was the Father of Christ? Matthew 1:20 says that the Holy Ghost was the Father, while Romans 15:6, II Corinthians 11:31, and Ephesians 1:3 say that God was the Father. There is no contradiction when we realize that God the Father and the Holy Ghost are one and the same Spirit. Matthew 10:20; Ephesians 4:4; I Corinthians 3:16.

56. Can Trinitarians show that three divine persons were present when Jesus was baptized by John? Absolutely not. The one, omnipresent God used three simultaneous manifestations. Only one divine person was present--Jesus Christ the Lord.