The vacuum quantum fluctuations themselves are very hard to measure directly, but some of their effects have actually been observed.
The Casimir effect is the most famous example. Take two parallel, uncharged metal plates, and place them very close to each other. They turn out to attract one another. The effect is explained by realizing that vacuum fluctuations of electromagnetic fields are restricted between the plates (like sound waves resonating in an organ pipe) but not outside of them. This causes a net force.
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