max output current on 74LS595 - how to read datasheet
There are two tables. The first table tells you that the high level output source current recommended range is up to 2.6mA and, when the output is pulling low, the current can be as high as 24mA into the output pin.

The 2nd table tells you that you won't quite get perfect logic levels under output load conditions. For instance, when sourcing 2.6mA the output voltage will only be typically 3.1V whereas when sinking 24mA the output will not be 0V but slightly higher at 0.35V typically.

These numbers apply to all outputs except \$\bar Q_H\$.

It's also worth noting that the voltage levels under load are when Vsupply is at its minimum value of 4.75V.

Also worthy of mention is that some devices specify this and that for output currents but if you tried to drive all outputs at that current it wouldn't work - always advisable to check what the maximum power supply current is for a particular chip if you're looking to driving LEDs. I couldn't see it mentioned on this chip so maybe you are OK.