Is there anything in 5e similar to the 'catch hold' in 4e when falling? We've been house ruling, but I can't find anything in the core books.
Unlike 4e, 5e does not allow a saving throw to avoid falling by going prone. This was likely in order to simplify the 5e rules (plus, 'Pushes' occur less frequently in 5e).
That said, 5e seems to encourage DM's calling for rolls whenever it seems appropriate. So if a PC falls off a ledge and asks if they could grab something on the way down, a DM might allow it (if it seems appropriate) using the Ability Check rules (DMG p.237). Literal cliff-hangers are certainly fun.
However, usually there was already some sort of roll made for the fall in the first place (like an attack roll, or a Dex saving throw to avoid a pit trap).