I am looking for data that compares the number of transmissions per month or year among the various digital modes such as DMR, D-star and Fusion.
I want to see how popular these technologies have been over time.
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I think this is an interesting question, but seems hard to answer. In the USA, perhaps you could FOIA the FCC to see what their monitoring stations cover and/or if they collect any data on this.
Otherwise, the digital modes you are asking about (DMR, D-star and Fusion) are typically used on the VHF/UHF bands. These transmissions do not propagate far, and so are by nature a "local" thing. Just sampling one local area would not necessarily be representative, either. I'm sure it varies between one group of users on one repeater vs. another repeater, and some wider regions will tend to different modes than others, and it likely varies significantly between, say, Europe vs. USA vs. South America vs. Asia…
Depending on the motive for your question, maybe a better approach or new followup question would be to see if radio sales numbers are available? Not that everyone uses the digital features, but it at least would reveal potential.
Another alternate approach might be to look in the repeater databases (or general radio system listings) and see which protocols are represented most.
Finally, if you primarily want to get a radio compatible with nearby contacts, see if there's an in-person general ham or repeater club or a social media group of hams in your city/district. If asked, I'm sure they'd be happy to offer some opinions of the best type to get based on what's popular locally.