SN74HC595N Shift register output bizare
Probably your switches are actually normally open (you chose the normally closed switch symbol), also there are a bunch of shorts between Vcc and SER, OUT4 and SW3, for example, that you probably do not intend. 

You need pull-down resistors to get predictable logic levels at the chip. Open on an HC logic chip input just means it can float around to some unpredictable state. Try 10K or 4.7K to ground. 







simulate this circuit &ndash; Schematic created using CircuitLab

You should have resistors in series with each LED to limit the current. 1K is fine. You should have a bypass capacitor (eg. 100nF) across the chip power supply. 

There is a more subtle issue- the mechanical switches "bounce" and you may (or may not) get many clock edges for a single press, so even with the resistors and bypass cap this will probably do strange-looking things, at least some of the time. 

Search for debouncing circuits to figure out how to add that. One of the simpler ways, assuming a SPST-NO switch, is to use a Schmitt trigger gate such as the 74HC14. 







simulate this circuit

You only need the debouncing on the CLOCK input, since you will only send a clock pulse when the data input is stable, and since multiple edges on the LATCH input will just transfer the same shift register data again.