I've looked all over the Semantic UI demos and couldn't find a Semantic UI equivalent to Bootstrap Popovers. These are popups that usually open connected to their toggle and can stay open for the user to interact with.
If no popovers exist in Semantic UI, what design / element do people recommend to use instead? The basic idea is to get something that works like the "Add new metric" button in this design.
As Aibrean said, there is already such a functionality, see Semantic Popup module.
<div class="ui button" data-title="Lorem ipsum" data-content="Lorem ipsum dolor sit amet." data-position="right center">Click Me</div>
<div class="ui button" data-html="<div class='ui secondary vertical pointing menu'><a class='active item' href='#'>Home</a><a class='item' href='#'>Messages</a><a class='item' href='#'>Friends</a></div>">Click here</div>
<script type="text/javascript">
$('.button').popup({
on: 'click',
closable: false
});
</script>
Here is working demo: http://jsfiddle.net/0q42ztp1/