We could query for patterns like
<script type="application/ld+json"> , eg:
#standardSQL SELECT COUNT(0) FROM `httparchive.response_bodies.2018_05_15_desktop` WHERE body LIKE '%<script type="application/ld+json">%'
The result is 88,467 pages containing the JSON-LD signature. There are obviously several other forms of structured data, but this gives us a rough idea.
Changing the table in the query to
2017_05_15_desktop (one year ago), the result is 25,471. So adoption of structured data is definitely growing, at least for JSON-LD!
With Lighthouse support in HTTP Archive, we will soon be able to more easily query for structured data usage and validity. There is a new audit being developed: https://github.com/GoogleChrome/lighthouse/issues/4359