Irony of "Twinkle, Twinkle, Little Star"
When Marvin goes Outside for the first time, he notices the stars, bringing to mind an old childhood rhyme his father had told him:
"Twinkle, twinkle, little star,
How I wonder what you are."
Marvin's treatment of his nursery rhyme is ironically rational considering his young age. He critically observes that the stars don't actually twinkle but give off fixed light, then thinks that the writer of the rhyme must be a moron because everybody knows what stars are! Besides being humorous, this segment illustrates a possible future society governed by scientific advancement and notes the irony of a well-known set of phrases.
Irony of Marvin's Dreams
Marvin has always dreamed of returning to Earth, and this dream manifests itself in smaller ways as well, such as desiring to go Outside the protective bubble of the Colony. These dreams, however, turn out to be impossible as Marvin realizes that he will never be able to return home, and his job is to make it possible for his descendants to do so.
Irony of Armageddon
Reflecting the fears and anxieties of his time, Clarke makes nuclear war the cause of the Earth's destruction in this short story. This sort of self-destruction is essentially ironic: rather than being destroyed by outside forces, humanity has delved too deeply into control over nature, and this ungodly power has led to a sort of ironic self-annihilation.
Irony of Psalm 37
The title of this short story is a reference to Psalm 37:5, where the psalmist laments for his native Jerusalem, which has been destroyed in Babylon's sack of Israel during the period of the Babylonian captivity. This parallel makes the story more poignant, certainly, and it brings a subtle hint of depression, but it also brings a hint of joy: just as Jerusalem was rebuilt, hopefully the Earth can be as well.
Irony of the Lunar Colony
Marvin astutely points out a peculiar irony of the lunar colony: because of the bleak landscape and lack of resources, the colony is safe from most dangers and disasters. This very dearth, however, can be the cause of another, equally detrimental danger: a lack of hope and motivation. Thus the colony, despite not being physically endangered, is still in danger of collapse from sheer lack of willpower.