Within the writings of defenders of the trichotomic nature of man there is significant variation on the question of the nature of angels. The trichotomy of man is a much more popular doctrine within Protestantism than in other branches of Christianity, so that's the perspective that I'll be able to share.
The simplest part of this question is, do angels have spirits? It's not surprising that all theologians I have found in this study hold that angels are spirit or have spirits. Evidence for this will be seen as I handle the remaining two questions: Do angels have bodies? Do angels have souls?
Do angels have bodies?
Defenders of the trichotomic nature of man fall into three camps on this question: (a) angels do not have bodies, (b) they have a particularly pure body, unlike our own, and (c) we don't know.
(A) J. Rodman Williams belongs to the first camp:
Angels are pure spiritual beings. [...] This is not an attribute or quality of their being; rather in essence angels are spirits. Angels, accordingly, are incorporeal: they have no bodies. A spirit, a pneuma, does not have flesh and bones.1
Another proponent of this view is Franz Delitzsch. He discusses the visible manifestations of angels as follows:
The angels have no bodies; but, by the miraculous power of their will, they might be able to make themselves visible, and to take what forms they pleased, according to the object of their mission, and the subjectivity of the beholder.2
This, he says, is the only sense in which it can be said that angels have bodies: that "like the divine nature, [their spiritual nature] is capable of manifestation."2
(B) The second camp holds that angels have bodies, but that these bodies are somehow more pure or more spiritual than those of man. Clarence Larkin simply states, "[a]ngels have bodies."3 John Bickford Heard, while saying that angels have no "physical nature,"4 nonetheless suggests that angels, like resurrected men, have "pneumatical bod[ies]":
[T]o man [belongs] the nutritive, sentient, and rational, with the germ of the spiritual, and to angels the sentient, rational, and spiritual, without the nutritive. [...] [S]ince man hereafter is to be equal with the angels, the nutritive or plant life cannot form part of his resurrection body. [...] This transformation of man from a psychical to a pneumatical body is analogous to the transformation of insects.5
(C) Those more hesitant to take a position on this question include Lewis Sperry Chafer, who writes, "[i]f the angels have bodies, their bodies are of a spiritual order."6 Similarly, Jan Jacob van Oosterzee calls the question "wholly uncertain" of whether angels were "created free from all corporeity, or simply from our more gross one."7
Do angels have souls?
Here we have more agreement, though some commentators do not directly address the question. Not surprisingly, members of the first group above, Williams and Delitzsch, do not believe that angels have souls. Delitzsch writes:
Not as though the angels also had souls: we could only speak of the souls of angels as of a [soul] of God. [...] Scripture only speaks of souls of men and of brutes.8
Heard does not speak of angels having souls, and Johann Tobias Beck writes that "[w]e never hear of a soul in angels."9 Chafer approvingly quotes10 another trichotomic theologian, Hans Martensen, who says:
It is precisely because the angels are only spirits, but not souls, that they cannot possess the same rich existence as man, whose soul is the point of union in which spirit and nature meet.11
In conclusion, we can summarize the views of trichotomic theologians as follows:
- Angels are, or have, spirits.
- Angels do not have bodies like men; if they do have bodies, they are in some sense spiritual bodies.
- Angels do not have souls.
- Williams, Renewal Theology, Volume 1, 173-74.
- Delitzsch, A System of Biblical Psychology, 81.
- Larkin, Rightly Dividing the Word, 93.
- Heard, The Tripartite Nature of Man, 87.
- Heard, The Tripartite Nature of Man, 354.
- Chafer, Major Bible Themes, Chapter 20.
- Oosterzee, Christian Dogmatics, 316.
- Delitzsch, A System of Biblical Psychology, 231.
- Beck, Outlines of Biblical Psychology, 9.
- Chafer, Systematic Theology, Volume 2, Chapter 2.
- Martensen, Christian Dogmatics, 133.