There are some write ups out there:
https://wexnermedical.osu.edu/blog/covid-19-vaccine-long-ter...
> Once the body creates that spike protein using the mRNA instructions, the body quickly breaks down those mRNA strands and they dissipate within a few hours or days after injection. The mRNA never enters the nucleus of any cell (where the DNA is located), it doesn’t affect any genetic material in the body, and the mRNA strands are removed from the body through everyday cellular processes.
https://www.nebraskamed.com/COVID/where-mrna-vaccines-and-sp...
> The Pfizer and Moderna vaccines work by introducing mRNA (messenger RNA) into your muscle cells. The cells make copies of the spike protein and the mRNA is quickly degraded (within a few days). The cell breaks the mRNA up into small harmless pieces. mRNA is very fragile; that's one reason why mRNA vaccines must be so carefully preserved at very low temperatures.
You can follow the links for fuller explanations.
If you want to see hard data, I'm also aware of two studies that investigate how long the spike-protein lingers after vaccination:
This one says five days: https://academic.oup.com/cid/article/74/4/715/6279075
And this one says ten: https://www.mdpi.com/1424-8220/21/17/5857/htm