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I installed those on my first three Town Cars and I have to say - drilling them is not what you're thinking. The very small holes are on the inside of the fender lip and cannot be seen. If you don't use the fender chrome that attaches with screws, be prepared to see them fall off/come loose after a few washes. I recently installed them on my 2004 Signature Series and was quite disappointed that I could not attach the rears with screws (there is no lip on the rear fenders after 1997). As for dirt and debris building up inside a drilled fender chrome - that can't happen. They fit flush to the fender, there is no gap to allow anything in if they are screwed on.
Sure it can happen. You get on the road in winter with sand and mud/slush flying everywhere and you'll get a healthy layer of grime behind those trims that gunks up from the bottom. With no way for water to run down from the top lip thanks to the rubber gasket edging, that soil and dirt can just sit there on the lip for however long it is until someone jams a hose up under and gives it a flush. I know this because I had them on my last 2 panthers, a CV and a MGM. I was careful to keep it clean under the trims because on those cars the front fenders are prone to rusting out and if you don't keep one eye on them at all times you're gonna end up replacing. So, to avoid this, everytime I washed either car I noted the grime and gave it a good spray and it would all run out in black streaks. Those were the drilled type. The clip-on type I've not used but I've been thinking about doing so since I like the look of trims but not the idea of drilling my currently pristine fender lips. Perhaps they will fall off, I don't know, but surely the hooks must do something. I'd keep a good check on the installation every now and again as well, just to be sure everything is seated nice and tight.
Here's a clip from youtube showing how the clips work:
http://youtu.be/m-BK20xFgbw
Here's a link to an eBay sale for trims that can be either drilled or mounted with clips provided:
90-94: http://www.ebay.com/itm/90-94-Lincoln-Town-Car-Chrome-Fender-Trim-/271079880189
95-97: http://www.ebay.com/itm/95-97-Lincoln-Town-Car-Chrome-Fender-Trim-/310710708292
Good luck!
That's a great YouTube video but the 2003-2011 Lincoln Town Car doesn't have a lip to bend the clip around. You can bend it up and ever so slightly closer to the inside fender lip only. And you won't get the snug fit depicted in the video. They do stay in place but you can easily move them around and I'm almost certain that a car wash would pull them right off.
2003? I was speaking about the 90-97 models just now since the photo posted by the OP was of that era. I can see it being a problem beyond that if there's nothing to bend the clips around though, like you say. I didn't search for the no-drill type for the 98+ TCs. Not sure if there's such a product even.