I've misplaced my user manuel for my Shark iron model G1408Z. Can you tell me how the self-cleaning works? Or how to properly clean my iron? It is leaving white residue from the steam, the shape of the iron, on my clothes.

Go on the internet and put in the model # and download the manual

I have provided a link to a manual that seems to belong to your model. But unsure what features your model has. This is not available for some reason at the Rowenta USA website.
Refer page 7 of the manual available at this site
http://laundry.manualsonline.com/manuals/mfg/rowenta/1103898336__4605.html?p=1

Self Clean - prolongs the life of the iron
The Self clean function flushes dirt and scale particles
out of the steam chamber.
To operate:
• Fill the tank with untreated tap water up to the maximum
mark and heat the iron to max. temperature.
• Unplug from the mains.
• Hold iron over a sink horizontally, push the steam control
3 past the position and into the Self Clean position
and hold it there: steam will now begin to build up.
After a few seconds water and steam will emerge from
the soleplate, rinsing dirt and scale particles out of the
steam chamber.

After approx. 1 minute move the steam control slide to
the position.
• Insert mains plug and allow the iron to heat up again.
• Wait until the remaining water has evaporated.
• Unplug from the mains and allow iron to fully cool
down.
• When cool, the soleplate can be wiped with a damp
cloth.
We recommend using the Self Clean function approx.
Every 2 weeks. If the water is very hard, the iron should
be cleaned weekly.
Self Clean Valve
The self clean valve prevents the drip hole into the steam
chamber from furring or blocking up. To ensure the iron
continues to function properly, open and close the valve
by sliding the steam control to max then min a few times
after use.

Just put it on the lowest setting that is near to the off position. Polyester can melt on a high heat setting.

Have you checked the fuse?

Try a "magic eraser". They're minimally abrasive. Then run the iron over a piece of waxed paper and then on a cotton cloth or rag.

Use Vinigar to clean the brown icky stuff off the plate.

http://www.sharkclean.com/europro/europro1/manual/GI477-GI477R_REV_0907_NO_PROP_65.pdf

OMG, how frustrating to be ironing and water shoot out everywhere from this iron....So, to remedy it, I have turned off the steam, use it as a dry iron, and have a spray bottle handy to spray as I iron...love the feel of this iron, but hate the water everywhere...

fill with water turn iron on max keep turning steam dial all the way to the right until you get to self clean hold iron horizontal and make sure you have a towel or something to catch water drips and debris coming out of iron let iron steam and spit water out until it stops on its own turn dial back to regular steam settings wipe residue off of the plate with a towel adjust temp settings back to normal

make sure you do a self clean as often as possible to remove debris and scale from iron