Despite all the bluster from the Ruth party, by all measures the Tories have been on a long, slow decline in Edinburgh, somewhere where they didn't do too badly in the leafier suburbs even 10 years ago (when they got 25% of the council election vote). Despite Ruth winning Central and despite her overall bloc of MSPs being fairly large, Edinburgh was one of their worst performing areas in the last Holyrood elections.
Local election PR changed a lot of things in Edinburgh; Labour used to get 50-60% of the council seats on 30-40% of the vote. SNP used to get 0 seats on 15-20% of the vote. The Tories seem to have always gotten a roughly proportional number of seats to vote share regardless of the system used. Probably because of their vote being more concentrated in a smaller number of wards than the SNP who would historically have a reasonable vote share but spread across all wards in the city and therefore not winning any individual seats.
The Ruth party is I think hoping to pick up disaffected Labour votes but I'm not sure what Brexit will have done for that. I wonder if the Lib Dems can capitalise on this and take some votes where they historically used to be strong (West of the city) from folks who can't or won't vote SNP. When the LD vote collapsed in 2012 (from 30% to 5%), it didn't give the Tories any sort of bump. Labour and the SNP got a reasonable bump from it (proportionally, the 2012 turnout was low enough that they had less votes than the previous election. Perhaps all the LD voters just stayed at home?)
Like the Holyrood elections there's likely to be some sort of anyone-but-the-SNP factional voting going on, e.g. Labour in the South, Lib Dems in the West, Tories in the New Town. The dynamics of this are made especially complicated and hard to predict by the transferrable vote system. I think this system also makes it fairly harder for the SNP to make the sort of universal sweeping gains they managed in the FPTP GE system.
I don't think Labour will get the spanking that some might predict in Edinburgh, I imagine they will lose seats but still hold up a reasonable vote share which with the PR system will allow them to maintain a fairly large bloc. Andrew and Lesley are retiring yes, but are fairly well thought of and don't seem to be as muddied by Labour factionalism as the MP/MSP contingent are. I could be entirely wrong on that of course, perhaps there will be Corynbotrot Entryism or a NeoBlairite Putsch to try and stuff the lists of Labour candidates. But will it be larger than the Tories?
Think the Greens should pick up a few more too, Alison Johnstone did well in Central at the Holyrood election. Andy Wightman well thought of too. Their councillors don't seem to have attracted the sort of scandal that some members of the SNP crowd have. A lot of Edinburgh voters do seem to have Green issues fairly high up their agenda, whether or not they actually vote Green.