Discourse by default does not support signatures. The official plugin that enables them is installed on here, however that they are not visible by default for user accounts created on the old vBulletin site. New accounts should have them enabled automatically, and they may be visible on some older accounts if you never made any settings or profile changes.
To enable them for your account you need to do the following:
Desktop Method
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Click the avatar menu in the upper right hand corner of your screen. It may just be the first letter of your username on a color background if you have not set one for yourself. It may also be your Gravatar if you have one set for your email address.
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Click on your username in the panel then Preferences
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Click Profile in the sidebar. No Sidebar? Follow the directions under On Mobile instead as it is likely your browser window is too small to show the Desktop UI and it auto-switched to the hybrid view.
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Under Enable Signatures Check the box next to See user signatures below posts
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(optional) If you wish to have a signature, fill it out in the My Signature section. Signatures accept most of the same formatting as posts.
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Click Save Changes at the bottom of the page.
Congratulations, you have successfully enabled the display of signatures and (optionally) set your own.
Mobile Method
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Tap the avatar menu
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Tap your username on the menu that pops up
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Tap Preferences
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You will now see two dropdown menus (they may be side-by-side or stacked depending on your device). In the right-hand (side-by-side) or bottom (stacked) menu, select Profile.
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Under Enable Signatures Check the box next to See user signatures below posts
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(optional) If you wish to have a signature, fill it out in the My Signature section. Signatures accept most of the same formatting as posts.
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Click Save Changes at the bottom of the page.
Congratulations, you have successfully enabled the display of signatures and (optionally) set your own.