Am I using this multiplexer (SN74LS153N) correctly?
The clearest problem is that U1 pin 1 (1G) is unconnected, and so it will typically float HIGH on TTL devices.

That means that U1 pin 7 (1Y) will stay LOW, irrespective of the states on the inputs U1 pins 3 - 6 (1C3 to 1C0).

See the first line of the truth table, in the datasheet which you kindly linked:





I understand that they are just for troubleshooting, but that U1 LS TTL outputs don't have enough source current to drive an LED (i.e. with the LED between a U1 output and Gnd) and stay within spec. The datasheet says the recommended maximum source current is just 0.8&nbsp;mA (800&nbsp;&mu;A).

Better to use a transistor (BJT or MOSFET) driver, to avoid damaging the U1 outputs and to still get a valid HIGH logic level for U2 to read.

This previous question explains the problem.
Finally, you should have decoupling capacitors close to U1 and U2.

You might get away without them, but that would depend on several factors and best practice is to include them.



This short Fairchild "Designing with TTL" Application Note 363) has some useful advice on various topics, including decoupling and what to do with unused TTL inputs.