It's sung/read at Matins in the EO.
Besides being read at Matins, as NicholasMyra noted, Psalm 50 is also one of the psalms of the Third Hour and is also recited by the deacon as he censes the church in preparation for the Divine Liturgy. Also, following the revelation of the angel to St. Pachomius, which is usually printed at the end of the Psalter (but not, alas, in most English psalters), Psalm 50 is part of the beginning sequence of Morning and Evening Prayers. Here is the revelation to St. Pachomius I am referring to, which is, no doubt, part of our shared patrimony with the Coptic Church:
FROM THE FATHERS CONCERNING THE CELL RULE
and about the Prayer which the Angel of the Lord gave to Pachomius the Great. 
A brother asked an elder, saying, I pray thee, Father, tell me. How should I spend my time in the cell? And the elder answered, I keep my own rule in this way: at night, I sleep four hours, and for four hours I stand in psalmody, and I work four hours; and during the day, again, I work until the sixth hour, and from the sixth I read, even until the ninth, and from the ninth, I cook my food, and that is how I do in the cell. Again the brother asked, But how many prayers should I say? And answering, the elder said, I do ac-cording to the revelation of the Angel, which he gave in writing to St. Pachomius. And you can do likewise. And the brother said, What is the Angel’s revelation, Father? And the elder said, It is recorded in the Angel’s written revelation, how the monks which were under the authority of St. Pachomius might do twelve prayers during the day, and twelve at night, and at the ninth hour three prayers, and to sing a psalm before each prayer. And he said this also, I started slowly, for it is good to keep the rule, even in part, and no one will be sorry. For perfection hath no need of regulation, as thou hast heard from him who said, Pray without ceasing.
The rule which the Angel of the Lord gave to Pachomius the Great.Begin with the
Trisagion. After the
Our Father: Lord, have mercy [12 times]. Glory, Both now: O come, let us worship, thrice
. Psalm 50, Have mercy upon me, O God
; I believe in one God; one hundred prayers
, O Lord Jesus Christ, Son of God, have mercy upon me, a sinner. And then
, It is truly meet…, and the Dismissal. And this is one prayer. It is commanded to perform twelve of these in the day, and twelve at night.
 See also St. John Cassian, On the Institutes of the Cœnobia
, Book II.