The tricky thing about answering a question such as this one is to recognize that, ultimately, Homer wrote in Ancient Greek. Therefore, unless you are working with the original Homeric text, you'll be working with one of a wide range of translations, and this is something which you need to take into account. No two translations are going to be identical, and this creates a wide range of variation in terms of the epithets you might find. Be aware, for this answer, I will be drawing from the Penguin Classics edition of The Odyssey, translated by Robert Fagles (New York: Penguin Books, 1996 (paperback edition)).
In any case, epithets did represent an important part of Greek epic literature, simply given its mnemonic basis. Epic poems such as The Odyssey had their origin in the oral tradition, and epithets tend to serve a critical function within this storytelling form. Thus, you should not be surprised to find numerous examples of epithets running across this poem.
For example, there is Athena, who is referred to with epithets such as "sparkling-eyed Athena" (Book 1, p. 79) and "bright-eyed Pallas" (book 2, p. 105). Then there is Zeus, who is referred to with epithets such as "Zeus who marshalls the thunderheads" (p. 79) or "farseeing Zeus" (p. 98). We see in book 3 a more complex use of epithets, by which Athena is referred to in reference to Zeus (who himself is attached with an epithet): "Athena, daughter of Zeus whose shield is storm and thunder" (p. 108).
Of course, also note that epithets are not solely invoked in reference to the gods. Telemachus is referred to with such epithets as "heedful Telemachus" (p. 87), "cool-headed Telemachus" (p. 90) and "self-possessed Telemachus" (p. 103), to name a few. You can also point towards Nestor, who is referred to as "Nestor the noble charioteer" (pp. 110, 115) and "Nestor, breaker of horses" (p. 108).
The Odyssey has innumerable epithets. They can be found in these first four books and across the entire length of the work in question.