My polaroid TLA-04011c wont come on at all - it has a blue LED lit, indicating the set is on, but theres no picture whatsoever

www.hdtvreview.com/pdf/polaroid-tla-04011c-manual.pdf
Above is your manual, you say there is a picture, but it will not "play" Check, you are connected to an antenna, or to an external video source, and the right input has been selected.

Does your set respond to the controls on the set itself?

If you, are you able to access the setup menu to perform a factory reset?

If so, press menu, navigate over to setup, then choose reset defaults, to see if that improves your situation.

I hope that helps.
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Polaroid TLA-04011C Manual Download
Polaroid TLA-04011C Service Manual Download

Look at the power supply first; look for burnt parts, bad capacitors, etc.
http://en.wikipedia.org/wiki/Capacitor_plague
Parts: http://www.shopjimmy.com/catalogsearch/result/?q=Polaroid+TLA-04011C+

Please post back what you find

The most likely the TV has a power supply related problem or it maybe just a blown fuse inside the TV.
I can't say it's 100% the problem because I can't actually look at the tv over the internet.
If by some chance the TV is still under warranty, go that route.
If not under warranty and If you have some electronic experience, you can purchase a Service Manual over the internet and diagnose the source of the problem and fix it by soldering in the new part. The Service Manual will give you circuit diagrams, test points, and part# lists. Usually, cost under $20.

I hope I shed some light on the subject for you.

Polaroid TLA-04011C

You can get a service manual from;
http://www.polaroidsupport.com/polaroid/docs/TLA-04011C_TLXB-4011_TLXB-4241_ServiceManual_20070613.pdf

This will give you diagnostic flowcharts etc. This could be useful in the future.

Check the powersupply/inverter board for bad capacitors. Also check the CCFL bulbs in the top and bottom for poor solder connection - replace if necessary.