How do I permanently delete all my inbox e-mails?

Top left of your Mail screen there is a left arrow. Click and it should show your mailboxes. Below are a greyrsion of the mailboxes. Click the down arrow on the mailbox you have junk mail in. It will open to show all your mailboxes for that account, inbox, sent, drafts and junk. Open junk and click edit. Each junk mail item will show a circle to the left. Click the circle and it is ticked. Click delete. Also you can delete the lot by clicking delete all, bottom right. Good luck.

They might be going to a spam folder or unsubscribe folder and not the inbox. If they are there move them to the inbox and i think next time thei will go to the inbox

How to delete or move ALL emails at once in non-jail broken ipad or iphone
It took me weeks of research to figure out finally how to decode this yet another secretive secret of apple. There is a perfect way of deleting ALL emails at once without jailbreaking your iphone or ipad...and here it is:
>> Open Inbox >> Edit >> Mark All >> Mark As Read [Added May 21, 2014 by Adrian; this extra step makes the difference between success & failure for some.]
>> Open Inbox >> Edit >> Check/select the top message; it will highlight the move button.
>> Press and hold the move button and uncheck the message that you had checked earlier.
>> Lift all your fingers off from the ipad screen and leave it alone. Wait until all your messages pile up on the right hand portion of the screen (in ipad); iphone will give you the actual number of emails it has selected for the action. Now they are just waiting for your command to be moved ALL at once.
>> Choose trash to delete all of them or any other folder where u want to move them, like one piece. Remember this will replicate your action on the server so you will ACTUALLY move them or delete them on the server and not just the ipad.
Once you have moved all messages to the trash you can either leave them there for the scheduled cleaning or empty it right away by doing this: go to trash folder and touch Edit. The Delete All button shows up at the bottom of the screen. Hit it! You're done!
If you do not see the effects of your actions on the server make sure you have enabled your email accounts for such actions.
Tips:
1. Please give enough time (could take up to several minutes depending the number of emails to be moved) for the emails to be selected for the move. Your screen will become unresponsive while all emails are being packaged. Once emails are ready, in ipad, you will see them zoomed out on the right hand side of the screen, and in iphone you will see the message showing you the actual number of messages that have been selected.
2. Avoid purging very large number of emails, the mail app might freeze or crash. If your inbox has thousands of emails change your sync settings in mail settings to fill lesser number of emails in your inbox.
3. [Added Jul 20, 2014 by Adrian] Many people have reported needing to repeat the above procedure several times before it succeeds. (I too have found this to be necessary a few times on my iPhone but not on my iPad-go figure.) So my final tip is to repeat the procedure 3-4 times if the mail doesn't disappear the first time. In my experience, if your messages disappear momentarily and then reappear, repeating the procedure will eventually make them stay deleted for good.
4. [Added Sep 18, 2014 by Adrian] If you are using IOS 8, there is a setting that may reduce emails piling up on your device! Go to Settings -> Messages and select Keep Messages under the Message History header. You'll be able to choose to keep your messages for 30 Days, 1 Year, or Forever.

Mail or iCloud (or gMail, if you use that, or in general the mail system) thinks the mail is junk, for some reason. You need to teach it that it isn't, by selecting the mail, press the folder icon and send the email to Inbox. Eventually, it will learn the difference.

When you're in Gmail, press the three little bars at the very top left edge to list your inbox and labels. Press Trash to go to that folder. Then press the three dots at the top *right* and choose Empty Trash from the dropdown menu. When it asks "Empty trash?" press Delete and you should get a message "No trash here. Thanks for recycling!"

Is it a virus and do you have anti virus protection? or sign in to your account and tick the select all box for these emails, then delete all.

Open your inbox and click on edit. When the small circles, click the msg you want to delete. Then click delete and the msg should into trash. You will still need to delete from trash and they should be gone.

Go to Settings>Mail, Contacts, Calendars. Under the Mail window - go to Show> and select the number of recent emails that you want the iPad to load in your email accounts.

While you are in the mail settings - tap on the email account, tap on the account in the next window, tap on Advanced in the next window and tap Delete - and select the amount of time that you want the emails to stay on the iPad.

In some mail accounts there will also be an option for Delete from Server - tap that and set it to never if you want to control when to delete your old emails.

Go back to your email inbox in the mail app and swipe down to the bottom of the window - there may be an option to "load more messages from the server" in blue letters. Tap that and you will download as many odl emails as you set up for in the Show recent messages setting. At the top of the inbox is a search field - type in keywords in that search field to find old emails.

You can also search for emails in the spotlight search. Go to the home screen and tap the home button and the search feature pops up. Type keywords in the search field at the top to find old emails there as well. When you tap on that email in the search, the mail app will open and you can read the email.

In mail, when you are in a specific mailbox, on the top right side of the left hand column (in portrait view) next to "inbox" press "edit". Then in the bottom left corner there is a "mark all" option. You can do that and the unselect what you want to keep.

The other option, if you don't like those red numbers over the mail app is to follow these directions:
settings > notifications center > mail > select the desired mail account if you have more than one and then turn off "badge app icon"
this works for any app that has those (sometimes annoying) red icons. If you don't want that large number of unread emails on your mail app, use this method. If you want to delete them, use the previous suggestion.

Hope this helps!

iPad 3- iOS 7.1.2
iPhone 5S- iOS 7.1.2
iMac 21.5"- late 2013

1.Create folders within your email program to store your emails. Make the folders appropriate for storing the emails you receive. For example, if you regularly receive emails from certain people, make a folder to store emails from each person. Make folders to store professional emails and personal emails, and make subfolders within each to further categorize and sort your emails.

2.Set your message rules to route your emails to your specific folders. Many email programs have different levels of junk email filtering. Choose a low or high setting based on the volume of junk mail you receive. Once you have created message rules and junk email filtering, the junk email will be easily noticed when it arrives in your main inbox. Your email program will also attempt to recognize junk mail and will send it to junk and spam folders.

3.Take a few moments each day to browse through your email folders to review the emails you receive. Important emails sometimes get routed to the wrong folders, and they can go unnoticed if you do not check your email folders regularly.

4.Open your spam or junk mail folder and make sure these folders do not contain any emails you wish to keep. If there are emails that are not junk, drag them to the inbox or another folder for storage. Depending on your email program, you also may highlight the email and find a "Not Junk" option to alert your program that this email does not belong in the spam folder. Once you are sure the folders contain only junk mail, highlight one email folder. Press "Control-A" to highlight all of the emails in the folder. After highlighting these emails, press the shift key at the same time that you press the "Delete" button. This will delete all of the emails without sending them to the deleted folder.

5.Manage your junk email filter by opening the "Options" in your email program. From here you can add "Safe Senders" or "Blocked Senders" to either list.