Citizen science is, in my completely unbiased opinion, awesome. It allows scientists to gather much larger, more geographically comprehensive amounts of data than they would otherwise be able to do without massive funding–while at the same time giving ordinary people the opportunity to learn about the world while contributing meaningfully to real science.
This page will collect information about all the citizen science opportunities I know of involving odonates (dragonflies and damselflies), because I think it would be nice to have that information in one place.
If you know of any other citizen science efforts involving dragonflies, particularly efforts not limited to North America in scope, please drop me a comment below so I can add them to this list!
Odonata Central (Global)
Don’t have the time to contribute regularly to a focused project? Odonata Central is dedicated to collecting information about “distribution, biogeography, biodiversity, and identification of Odonata (dragonflies and damselflies) world-wide.” You can upload the occasional sighting or add sightings regularly. OC requires you to have either a photo voucher or a collected specimen (I would avoid the latter unless you’re already knowledgeable about collecting, in which case you probably do not need this article) so recognized experts can validate your sighting. Odonata Central is also global in scope.
Migratory Dragonfly Partnership (North America)
A spinoff of Odonata Central, the Migratory Dragonfly Partnership hopes to shed light on the fascinating but surprisingly little-understood phenomenon of dragonfly migration. The MDP currently has several initiatives, which depending on your geographic location in North America could involve 1) monitoring life events of Common Green Darners (Anax junius) and Black Saddlebags (Tramea lacerata) at a pond you return to regularly, 2) collecting exuviae (the remains after an adult emerges) and adult Green Darners for stable isotope analysis, 3) monitoring the migration of five common migratory species in North America.
Dragonfly Swarm Project (Global)
If you happen to be lucky enough to sight a dragonfly swarm, consider reporting your sighting to aquatic entomologist Chris Goforth. Dragonfly swarms are a relatively common but poorly-understood phenomenon, and the nature of swarms is such that citizen science is probably the only practical way to collect data–so your observation is vital.
USA National Phenology Network (USA)
The USA National Phenology Network seeks to “monitor the impacts of climate change on plants and animals in the United States.” Phenology is simply a fancy term for the study of life events, such as hatching, metamorphosis, or mating. While not strictly dragonfly-focused, NPN does include eight dragonfly and damselfly species on its observation list, and provides datasheets for collecting information.
Nature Odes 
Dragonflies of India http://indiabiodiversity.org/group/dragonflies_of_india/show
Seeks to crowd source observational data to curate spatial, temporal and descriptive data on the odonata species of India
A belated thank you for the information, Thomas! I’ll edit the list to add it when I get a chance. :-)