How to enable push notifications of emails? A quick question with perhaps no answer,

How can I enable push notifications of emails?

What currently happens when I get an email is that my phone vibrates or rings, then I look at the locked screen and no information is displayed.

I'd like a preview of the email in a notification window with at least the sender.

I currently have 3 mailboxes, one in google, one in hotmail, the other a private work exchange server.

Is there a solution to receive push notifications for all of my mailboxes?
You're talking about the notification settings for Mail, not whether push is on. To see new mail messages in the Lock Screen: 

Settings > Notifications > Mail > Make sure "View in Lock Screen" is turned ON. 

If you're concerned about your mail's push settings, those are in:

Settings > Mail, Contacts, Calendar > Fetch New Data > Make sure "Push" is set how you want it.