Thanks all, I tried adding better decoupling for the 555 and shift register and made a shorter jump to get to the clock input. I got the rise time down to ~150nS, but still see the issue. I'm using the ON Semi part which has the max rise time as 500nS.

I'm questioning whether the issue is with double counting the clock now. I've attached some plots of me messing around with the input. Ch1 is clock and Ch2 - Ch4 are outputs 1-3.

Most times, instead of just putting the input onto the first bit, both the first and the second go high with the third behaving normally. Bringing the input low shut all of them off on the next clock cycle. Sometimes it seems to behave normally like in attachment 4 where the high input propagates like it should. Unfortunately, I don't have a logic analyzer, so I can't show more bits than that.
@garethw, I checked and don't have any floating inputs. The only pins I left unconnected are outputs 4-8 (and I suppose 1-3 which are just going to the oscilloscope).