Live.at (Outlook.com) doesn't allow you to link or share messages via their webmail interface, but they support IMAP, which means you can use other email programs like Mailspring to check your mail and create shareable links from email threads.
Follow these steps to get a shareable link for an email thread in your Live.at account:
Mailspring is a free desktop application for checking your email that makes it easy to share emails. It also comes with other great features like customizable themes, keyboard shortcuts, and read receipts! It works great on Mac, Linux, and Windows.
Run Mailspring and choose IMAP / SMTP on the email setup screen. Enter the connection settings for your Live.at account and click "Continue". Mailspring is a full email client and allows you to check your Live.at (Outlook.com) email without using the webmail interface. Your email should be displayed in a few minutes.
When viewing a message in Mailspring, click the Share Icon to open the sharing panel and click the "Share this thread" checkbox. Copy the link that appears and share it!
Anyone with the link can view the conversation and download attachments. If new emails arrive in the conversation, Mailspring will add them to the shared link. You can stop sharing a conversation at any time by unchecking the "Share this thread" checkbox.
Click your email provider below to learn how to connect to access your account with IMAP: