In Thunderbird, use CTRL+A to sellect them all and hit the Del key. It can take a while to finish.
You can also sellect all messages older than a certain age. Lessect the LAST message to delete. Scroll to the FIRST message to delete. Hold the SHIFT key and click the first message. This sellect all messages up to the last one to delete. Hit the Del key.
It can be similar with other clients.
You can also delete those files to delete EVERYTHING from your inbox:
The parts in brackets are specific to your installation.
In the folder: C:\Users\\AppData\Roaming\Thunderbird\Profiles\.default\Mail\
Delete the files inbox and inbox.msf files.
On a web-mail service, just don't connect long enough (usualy 2 to 3 months) and the account will be deleted, suppressing all messages.
Electro, November 2013