Do keep an eye on the Guild of One Name Studies Histories section where snapshots of revisions to the yDNA BigY DNA grouping results are posted every so often, the latest being as at Dec 2022.
We are gradually increasing our knowledge of which lines are more closely related.
Any Fairbairn who has DNA tested is welcome to join the FamilyTreeDNA Fairbairn Surname Project.
You can upload your test from Ancestry, MyHeritage, 23andme into FamilyTreeDNA for free. There is a small one-off free to unlock the additional tools beyond the match lists and contact info, such as FamilyMatching to split your matches by "side", the chromosome browser ++.
If you are a direct male line descendant, please use the contact details for the project to ask the admin (me, Lorna) whether or not your line is already represented with a yDNA test, or needs another to confirm the DNA signature. Subsidies may exist, depending on the answer to this!
Start at y37 to get a toe in the water and see who all pops up as matches.
Enjoy the journey