The most comprehensive source is probably the Oneirorcritica by Artemidorus Daldianus, a professional diviner and interpreter of dreams in the 2nd century AD, within the greek-speaking part of the Roman Empire.

While this is a treatise on dream interpretation, it's contents are fairly extensive and from it we can discern that his clientele dreamt about a huge range of subjects.

He provides plenty of oddly specific examples: being harnessed to a cart like a four-footed animal, writing backwards, walking on the sea, that stilts have been bound to one's feet.

As well as the mundane: washing clothes, seeing a midwife, drinking cold water, wearing a garland of flowers. Dreams of animals, mud, playing the harp, or dreams relating to a variety of other mundane objects.

Some dreams are frightening or horrific: eating human flesh, eating one's own flesh, being strangled or hanged, being crucified.

Some relate to gods, heroes, demons or the supernatural, going into some detail about the significance of each god, including lesser gods. Other examples in this category include descending into Hades, if serpents change into men, dreams of flying.

Others relate to family members, professions

Nor does he demur from discussing sexual dreams - including the bizarre: intercourse with oneself, the gods, with the moon.

While the interpretations are generally metaphorical, their meaning is not fixed, and often seem to have a psychological basis - their interpretation almost always depending on the circumstances of the individual dreamer:

To give a lively example:

'If a childless man dreams that he is kissing his own penis, he will have children. If he who does so has children who are in a foreign country, he will live to see their return and will kiss them. Furthermore, many who were single have married after this dream. If a man dreams that he has practiced fellatio with himself, it is auspicious, provided he is a poor man, a slave, or a debtor. For they will be released of their necessity. But it is bad for a father or a man who wishes to be a father. For, in the case of the former, his children will die: in the case of the latter, they will not be born. For the penis corresponds to children, and the mouth to the grave, since the mouth destroys and does not preserve what it receives. This dream predicts, moreover, the death of a wife or mistress. For a man who can furnish himself with sexual pleasure has no need for a woman. For other men it prophesises either grave financial difficulties or illness. For they have to resort to the "bare necessities" in order to stay alive, that is, they will sell whatever they do not want, or their bodies will be consumed by illness so that they will be able to bring their mouths to their genitals because of their emaciated state.'

Sometimes Artemidorus speaks generally of the meaning of particular dreams, but often he cites specific dreams attributed to an individual:

'A "cyclic" or choral flautist dreamt that the soles of his feet were eaten by worms. He retired and stopped playing the flute and competing in contests. In this way his feet were as inactive as if they had been eaten up by worms, since he no longer trod the stage as a performer"

"A man dreamt that he had three penises. He was a slave at the time and was set free. He then had three names instead of one, since he acquired an additional two names from the man who set him free"

"A man who was living in Rome dreamt that he flew around the city near the rooftops and that he was elated by his adept flying. All those who looked at him were struck by admiration. But as a result of a certain pain and palpitations of the heart, he stopped flying and hid in shame. An extraordinary man and an excellent prophet, he lived a distinguished life in the city. He not only gained people's admiration but a great fortune as well. But he did not get to enjoy either his prophetic art or his financial gain. For his wife no longer loved him and was unfaithful. Consequently, he left the city in shame."

"A man dreamt that he had an iron penis. He fathered a son who killed him. For iron is consumed by the rust the it produces from itself."

"A man dreamt that an olive tree was growing out of his head. He devoted himself zealously to the study of philosophy, both to the teachings and the way of life that it entails. For the olive tree is evergreen, firm, and sacred to Athena. And the goddess is thought to be the embodiment of wisdom"

"A shipowner dreamt that he was on the Island of the Blest and that he was restrained by the heroes until Agamemnon came and released him. He was impressed for public service and was seized by the procurators of the emperor. Later he was brought before the emporer himself who released him from his public obligation."

"A man dreamt that his slave, whom he valued beyond all the others, turned into a torch [the kind which is a lantern]. The man went blind. This slave guided him by the hand and, in this way, he saw the light through him."

Tl:dr people within the Roman Empire dreamt of a vast range of things as we do today, some of them culturally specific or peculiar to their religious/cultural/supernatural beliefs.

Source: The Interpretation of Dreams: Oneirocritica; by Artemidorus. Translation and commentary by Robert J White, 1975.