We need to improve the archaic style of the tray menu. It has a lot of text entries and misses focus.
When you look at Dropbox and Google Drive, both show a list of last uploaded files with a quick option to share those files.
cc @camilasan @jpnurmi
Some early mockups on an improved tray menu:
The goal is to achieve a better information hierarchy than just with the flat menu as we have now. Obvious inspiration are the new Dropbox tray menu and CloudApp.

Nice! I have never actually tried customizing a tray icon menu like that, but it might be possible to implement it as a custom popup if Qt's menu is not flexible enough.
- The menu should show regardless whether the tray icon is clicked with the left or right mouse button.
Now LMB shows the application. How would you do that otherwise. Not in the menu, please ;)
Nice. There should also be a place to show the shortcuts to the server apps. Same feature as exists at the moment already
It could definitely be extremely appealing to have an elegant tray menu, but for me the single most helpful change to the tray interface would be the ability to open the main window merely by double clicking, rather than selecting Settings, which is located at bottom of the drop-down menu, and which is really status and settings combined.
I also note a change in the e0b32c1 release, compared to the ef9315c, in which the tray icon is colored, rather than being grayscale, as it had been on my system before, which I had assumed was a selection made to respect my desktop theme.
As I am sure also has been considered, an icon ideally indicates the function or branding of an application, rather than being simply a check or cross to indicate status, as they might apply equally to any function.
Returning to the actual intent of this issue, a nicer menu might be very handy, but I would only caution against a design in which a menu expands to arbitrary size because of long file names. These scenarios impair usability and in many cases evade the notice of testers.
Double clicking on tray icon to open the main/first folder is pretty simple to achieve and should be implemented as an option if user don't like this behavior.
This is the default behavior for Dropbox (Windows/Linux) and Drive (Windows).
c.f. : #1184
Double clicking on tray icon to open the main/first folder
For me, it would be preferred to be able to open what is currently called the Settings window by double clicking.
Running under Linux Cinnamon desktop, I find that double clicking simply drops down, then pulls up the menu, each of the actions corresponding to one of the clicks. The double click is not given any special recognition, which is a waste.
Users may vary in preference for which action to map to a double click, but it definitely seems that some action should be possible. For me, again, opening the Settings window is the most natural choice since the folders reside in the file tree and are accessible independent of NC client.
As mentioned there is an issue open for that already at https://github.com/nextcloud/desktop/issues/1184 and it’s separate from the topic of this here, so let’s not clutter up the comments. Thanks :)
I can open a new issue if you want, but #1184 does not cover my comment. The author of the other issue states (emphasis added):
Single click could still be left as opening the app status/settings, but the double click captured to open the folder.
On my system double clicking causes no specific or helpful action. I observe that single clicking opens (or closes) the drop down menu, and double clicking is processed as two single clicks. Opening the settings window is impossible except through using the menu entry, which is inconvenient as it is by far the most common action.
Perhaps it is a platform issue.
Would you like me to open a new issue?
@brainchild0 please add your concern in that issue, but not here. :) Thanks
Current (open) PR on the share dialog: #1429
Updated mockup as of 2019-10-26:

Some explanation for the mockup:
Another update only regarding the heading as 3 dropdowns in the tray menu are a bit much, and the account switcher and the "Pause account sync" and "Log out" is also very related but at different places.

Combined this time. :)

With links to browser versions:

Current progress can be followed here https://github.com/nextcloud/desktop/pull/1565
Needs to be stabilized (see related PRs) but basically done.
👏 Great work everyone! :) Special thanks to @DominiqueFuchs!
Most helpful comment
Nice! I have never actually tried customizing a tray icon menu like that, but it might be possible to implement it as a custom popup if Qt's menu is not flexible enough.