I've had a lot of problems with icedove. It worked for multiple email addresses, then just one, now none!. I haven't worked too hard at fixing it, since there are so many alternatives.
I've used openmailbox with: pine, mutt, evolution, claws-mail, icedove, and geary. I could access it with all of those clients at one point or another.
My solution for now is I use one email client per email address. I'm using geary or evolution for gmail and I'm using claws-mail for openmailbox.
Geary is my favorite as it is clean and simple. However, it crashes randomly unfortunately--segmentation fault. Even when it works, it is a little to light for my taste--I want just a few more features. Lately I'm using geary in XFCE and it hasn't crashed yet. It did crash in several other window managers (fluxbox, i3wm, trisquel)
Several times I've considered getting a different email account because of all of this, and I may still. However, I've already used this email address in several important places and would rather not have to change it.
Has anybody had perfect success with the "web clients"? If not, what problems does it have.
