The most instantly noticeable group of formulaic expressions are the ‘heroic epithets.’ Each character within the epic has a set of epithets that are associated with him or her. Thus, Athene is ‘bright-eyed,’ Odysseus is ‘much-devising,’ and so on.
Open up your copy of the Odyssey at the beginning of Book IV, and list the epithets given to Menelaus within the book. You will find that he is variously described as ‘glorious Menelaus,’ ‘divinely-nourished Menelaus,’ ‘fair-haired Menelaus,’ and as ‘Menelaus, son of Atreus.’ Combinations of these epithets are sometimes used, and ‘leader of hosts’ is sometimes added on at the end.
The choice of epithet does not depend on the narrative context, but on metrical expediency. If you have access to a Greek text, write down and scan the Greek for each of the above epithets. You will find that they are all metrically different.
The Odyssey is made up of dactylic hexameters. What this means is that each line is made up of six feet. Each of the first four feet may be either a spondee (two long syllables) or a dactyl (long-short-short). The fifth foot must be a dactyl, and the final foot must be either a spondee, or a trochee (long-short or long-long).
The formulaic epithets allow the poet great freedom in the first few feet of a line that contains a significant noun, whether it be the name of a person or place, or even the sea or a house or a ship, since any space left over within that hexameter can be filled by a metrically appropriate epithet or combination of epithets, without any creative effort on the part of the poet. Furthermore, the use of formulae allows the poet time to consider the composition of the next line.
Similar benefits are conveyed by the frequent repetition of entire lines, and indeed of complete speeches. One of the more extreme examples of this within the Homeric corpus is Agamemnon’s speech in Iliad IX, and its verbatim repetition by the embassy to Achilles.