But if you want to blur a whole element (eg, your main page content while a modal is open) instead of just the part behind another one, that would be "filter: blur(<length>)" - and works in all modern browsers except IE (even my Android Firefox displays the demos on this page correctly):
...but only with a prefix, and only the most recent versions, and it doesn't have much market share. "Effectively nothing supports it", if you'd rather.
"Doesn't have much market share" really depends I guess on your domain doesn't it? For example, if you're building something that's going to get a fair bit of mobile traffic iOS Safari would represent a fair chunk.
Besides, most applications of backdrop-filter would fit into 'progressive enhancement' and the rest of the browsers that don't support it would just fall back no a non-blurred transparent background which is fine most of the time.
Yup. I noticed Apple has updated the top navbar on the new product pages where it is enabled. It falls back to unblurred-translucent container on other browsers.
Basically just Safari and only with a prefix.