Linking 1 output to 3 locations

This is a 'best practice' question.

I have a circuit with 2 multiplexers and a 595 feeding them. At the appropriate moment that I want to send data, I want to set the enable pin on the 595 and set the signal pin on the multiplexers to high (permanently). Is it an electronic problem to use one Arduino pin connected to all 3 of those at the same time?

An Arduino typically can drive 3 logic inputs.
But I don't see how it makes sense - which is the 3rd pin?
Can you provide a circuit diagram?

A important term here is fan-out. This indicates how many inputs the output can safely drive. In the case of the arduino you'll be fine.

However, you may want to be careful depending on the timing requirements of your circuit. If the multiplexer output has to be read immediately after the 595 is activated, you may get strange results. Another relevant term to study here is setup and hold time as this could affect whatever is downstream and reading the mux output.

I've got one 'signal' pin on each of 2 multiplexers, and one 'enable' pin on the 595 > 3 pins.

Thanks - yes that is a thing I considered but in my particular case there is no speed constraint. I can activiate, wait 500ms (or 5000ms) and then process each line in turn as fast or as slow as I care to. There are 25 x 25 checks to be done, so I'm expecting to let it run for a couple of seconds or so.

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