The Jan 1989 Ensign, page 29 tells that:

The Bible gives little detail about the personage of the Holy Ghost. The Prophet Joseph, however, gave us a number of insights about that Spirit Being and His office. On several occasions, especially in Nauvoo in 1842–43, the Prophet spoke of the Holy Ghost as a Being “in the form of a personage,” as a “spirit without tabernacle,” separate and distinct from the personages of the Father and the Son. According to the George Laub journal, on another occasion Joseph taught that “the Holy Ghost is yet a spiritual body and waiting to take to himself a body.”

Elder Bruce R. McConkie has said, "Spiritual bodies are tangible bodies, such as Christ had following His resurrection as found in Luke 24:36-43 and have spirit rather than blood. They are not to be confused with spirit bodies.

Heber C. Kimball taught: “Well, let me tell you, the Holy Ghost is a man; he is one of the sons of our Father and our God; and he is that man that stood next to Jesus Christ, just as I stand by Brother Brigham” (JD 5:179).

Franklin D. Richards quoted Joseph Smith as saying that “the Holy Ghost is now in a state of probation which, if he should perform in righteousness, he may pass through the same or a similar course of things that the Son has” (Andrew F. Ehat and Lydon W. Cook The Words of Joseph Smith, page 245).

BYU Professor Rodney Turner taught: There is yet much to be revealed in this world of the things of God, and even more in the eternity to come… (about this and other subjects.)

So, will the Holy Ghost ever get a body? From the above quotes it seems probable for His eternal progression, however, Brigham Young’s promise is that “when we go through the veil we shall know much more about these matters than we now do.” (JD 8:179)