Alliteration is used frequently in "Digging." The first example is in the first stanza:
Between my finger and my thumb
The squat pen rests; snug as a gun.
Here, the alliteration is working alongside consonance and assonance to create a "snug" sounding stanza. That is, all the sounds themselves seem to fit perfectly together, just as the pen fits right in the speaker's hand (and the spade in his father's).
The second stanza also uses alliteration:
... a clean rasping sound
When the spade sinks into gravelly ground:
My father, digging. I look down
The alliteration here is used to convey the physical effort of digging a spade into the earth. The prominence of the alliterating syllables draws the reader's attention to speaking as a kind of physical effort, involving the muscles of the mouth (indeed, even reading silently involves some muscle movement too).
The fourth stanza uses alliteration to a similar effect:
He rooted out tall tops, buried the bright edge deep
To scatter new potatoes that we picked,
Loving their cool hardness in our hands.
Again, these syllables have a sort of firmness or toughness that mirrors the effort of digging. The plosive, air-stopping consonants—the /t/, /b/, and /p/—are followed by the breathy exhale of the /h/, evoking, via sound, the physical intensity of this work. It's also as though the speaker is himself digging into the language, bringing up crops of consonance.
Next up is the /d/ sound in lines 23 and 24. Here "down and down" alliterates with "Digging," linking these words together conceptually. That is, by digging, the person doing the digging goes further and further into the earth. This also carries the same meaning as in the previous two examples.
The greatest concentration of alliteration of saved for lines 25 and 26):
The cold smell of potato mould, the squelch and slap
Of soggy peat, the curt cuts of an edge
The onomatopoeic quality of these lines is covered in the corresponding section of this guide. But it's worth noting how the numerous /s/ sounds seem to convey the dampness of the earth in which the speaker's father (and grandfather) would dig.