Ah! Finally, a question I can answer.
There has been some pretty good work about this recently. In particular, I would reccomend this paper by Chen et al. , which groups cyanobacteria into terrestrial / freshwater / marine and...well, the title says the rest. Some are edge cases by necessity (e.g. some coastal biofilm-forming cyanobacteria may be more similar to terrestrial ones than to many marine ones) but that's just nature.
Chen, MY., Teng, WK., Zhao, L. et al. Comparative genomics reveals insights into cyanobacterial evolution and habitat adaptation. ISME J 15, 211–227 (2021). https://doi.org/10.1038/s41396-020-00775-z (https://rdcu.be/dmJpm)
I regularly use this in my work, mainly because they have their trees posted on iTOL (in the 'Data availability' section) and this makes it really easy to search for specific strains.
Which brings us the the main problem. Right now there are about three main ways of separating the cyanobacterial phylum into orders. There is the NCBI database classification (largely based on 16S rRNA), like for example the Chen et al. paper uses. This may sometimes appear polyphyletic when overimposed on a genome tree. Then there is the GTDB classification (https://gtdb.ecogenomic.org/), which is based on genomes. This can be seen in Alcorta et al. 2020 (Alcorta J, Alarcón-Schumacher T, Salgado O and Díez B (2020) Taxonomic Novelty and Distinctive Genomic Features of Hot Spring Cyanobacteria. Front. Genet. 11:568223. doi: 10.3389/fgene.2020.568223) - on average, the tree shape is similar to the one in Chen et al., but the orders make more sense. The GTDB classification is good phylogenetically, and well-integrated with the database (for example, you can just get the list of all genomes/metagenomes available for an order). But because it's not focused on the taxonomy (the naming of things), the actual names may not always fit with the Bacterial/Botanical Code. One research group has been kind enough to go through the absolute mess that is cyanobacteria taxonomy, and now this paper exists.
Strunecký, O., Ivanova, A. P., & Mareš, J. (2023). An updated classification of cyanobacterial orders and families based on phylogenomic and polyphasic analysis. Journal of Phycology, 59(1), 12-51. (https://onlinelibrary.wiley.com/share/A5UXIHBUCIH9JRTIWFSE?target=10.1111/jpy.13304)
As far as I've seen, their orders more or less match with GTDB, just under different names (except for GTDB Cyanobacteriales with is split into more orders in this paper). They also have a much larger 16S rrNA phylogeny constrained by the genome one.
Hower, that solves only part of the problem. Because at the smaller scale (genus, species), you still have strains labeled by their morphologies, and culture collections are (for understandable reasons) hesitant to change their naming system. For particularly visually non-descript groups like 'Synechococcus' or 'Leptolyngbya', there is a good chance that a particular strain could be nearly anywhere in the phylogenetic tree. I would always get a 16S rRNA (if not already avaiable), BLAST it, and then check where related strains appear in a genome tree.
Edit: I have also heard of Cydrasil (https://www.cydrasil.org/), an online tool which can place a 16S rrNA sequence within the context of the cyanobacterial tree, but I haven't tested it yet.
Hope that helps!