Don't often make use of Webmail - mainly when holidaying abroad - so the Inbox gets pretty crowded. In the past I've been able to do a mass delete, or at least to set a time limit on incoming mail before it self-deletes, but can't find that option now.
To empty your Inbox, login to webmail, select Personal Settings then the Folders tab. Click Inbox to highlight it, then click the gearwheel icon at the bottom and select Empty. That will bring up a message box seeking confirmation. If you confirm I think the deleted messages will be moved to the Deleted Items folder and you would need to empty that too to completely delete them.