Arduino board not recognized by Linux (dmesg, lsusb, etc.)
So, to summarise the comments trail; the blinking lights at power-up are the normal boot sequence for the Leonardo,  indicating the processor is good, and that power is reaching the board; the device seen should be /dev/ttyACM0 or some other number; dmesg wasn't recording a plug-in event as the cable was not good.

We still don't quite know why the IDE thought it had sent a program to /dev/ttyS0, but presumably the level of interaction is much less (ie, none) via RS232 serial? Which explains why it said it had uploaded a test program when, in fact, it hadn't!

And replacing the USB cable has solved the problem - excellent news!