Why would you want to mount LEDs on the copper side of a single sided board? How about if you want to mount a board directly behind a front panel with the LEDs showing through holes in the panel. You can't put them on the usual side because the other components would stop the board fitting flush to the front panel, and you can't solder to the component side because there's no track there (unless you have a plated-through hole PCB). I have just such a project in mind and this is the solution I came up with.
All you need is a couple of strands from a multi-strand cable; A power cable from a PC strips down nicely. Put the LED in position and wrap a couple of turns of wire strand around one leg, pushing it down to the PCB with needle-nose pliers or a scalpel. Feed one free end through the adjacent hole in line with the track and cut off the other end. Solder the wrapped strand to the LED with a fine-tipped iron - See picture 3.
Turning the board over, hold the wire strand in place with a finger along the line of the track and solder as close as you can to the LED, being careful not to solder over the next hole along. Cut off the loose end - See picture 4. Repeat this and the previous step for the other leg and any other LEDs.
You now have LEDs mounted on the 'wrong' side of the board, and can continue adding the other components to the 'right' side. Drill the front panel to accept the LEDs using a step drill and a scrap piece of matrix board as a measuring guide
. Before you position the board on a metal front panel, make sure that nothing from the PCB will short out. Make a card spacer drilled the same as the front panel and put over the LEDs as an insulator before securing the PCB.
The final photos show the actual project build and how tightly you can pack LEDs with this method. There's 7 x 5mm LEDs in a 3/4" (2cm) square which will have additional components filling the rest of the PCB and be mounted directly into an Altoids Smalls box lid.