A 4017 and a 555 is also a solution.
Even if you use the 74LS95 you will need a clock source.
The issue with the 74LS95 is that you want to load just a single bit into it and allow it to circulate.
Connecting Q1 to P0, Q2 to P1, Q3 to P2, and Q0 to P3 will allow the "load" function to rotate the bits. That also doesn't give you a way to inject a single "1".
If you place an inverter in just one of these links (say the Q0 to P3 link) then the pattern will become:
0000
0001
0011
0111
1111
1110
1100
1000
and back to 0000
All this requires is a reset on power on.
If you also use a 74LS14, you have 6 Schmitt trigger inverters. One can be used to make an oscillator (I'm not sure how well this works with LSTTL), one can be the inverter you need, and the other 4 can buffer your outputs Q0 to Q3 to allow you to drive LEDs without affecting the logic levels. The reset can probably be done with an RC delay.