Like VK, I dislike the way GMail handles sets of messages (keeping track by Subject line) - so you cannot, for example, delete one item from a set of messages, or simply highlight a single message within a collection.
Don't get me wrong - I do use GMail and have at least 8 accounts (and there's a Firefox utility which can scan them at intervals so you are alerted to new mail arriving) and it's pretty good that it's free (there's another FF add-on which wipes out the advertising section so I only see my e-mail, not the side-bar of advert links too).
For paid-for services, since you didn't specify 'free' webmail:
I've used FastMail and Runbox.com for years, and tested MailSnare (now changed name and not tried it since), and they have some features I prefer over GMail/Googlemail (only googlemail if you register from a UK IP address), but both of them have annual fees.
Runbox also offers domain mail handling and web hosting as part of the deal, so if you have your own domain name like my sister hilary @ surname-family.info then it can accept all incoming mail for 'surname-family' and store in different folders, send alert messages to other mail accounts (such as Orange so she gets an SMS to show new mail arrived) and can even 'pull' in mail from other mail services.
Pretty sure you can add domain handling to Fastmail (I have one 'member' account and 2 old 'guest' accounts which are no longer available, so I don't pay any annual fee, but they have 3 or 4 levels of account, from memory).