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I've used the palm router on a few box-making projects, mostly the flush trimming bit. Really pleased with the results. I do approach it with a huge amount of respect though. It still amazes me that something so incredibly dangerous can be sold to pretty much any idiot who hasn't done their homework, such as myself. When I have more space I might get a router table so I at least place some bounds on the damage it can do.
Sharkstar-
Not to put you off, but a router table is arguably more dangerous because you're more likely to have your fingers near the cutter.
Spindle moulders (a very big router table for doing mouldings) are notorious for taking peoples fingers off!
I would also say that a router is a fantastically versatile tool, and once you know how to use one, they become indispensable.
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Flush trimming is definitely the fun end of routing. I have a little Festool one for trimming veneers which is easy work. Any kind of guided routing where the router is well supported can be quite relaxing.
I like to watch some good old fashioned safety videos every so often to get a visual image of how fast things can go wrong. I did rout the end of my finger with a Festool domino and was lucky to be within 5 minutes walk to Chelsea Westminster hospital where a plastic surgeon specialising in hand surgery was training, he came to A&E and made sure I could still play the guitar.
Nef
Airhead
That does seem like a common experience with routers. One of the important things to consider is shank size. If you decide you like routing and want to make fielded panel doors or using a template to cut worktops you'll need the larger size shank for the bits.
The smaller routers are so much easier for bearing guided edge routing though so most professionals end up with a large one and a small one, then possibly a palm one on top.
Frightening vibration and enormous amounts of dust are 2 of the down sides though. I wouldn't use it as a pillar drill, they tend to wander very quickly for all the reasons already mentioned.
I remember the days when I had a trend catalogue behind the cushion of the sofa and used to dream of all the projects I could get up to.