What's your favorite through-hole desoldering tool?

Yes, the wiggling does the magic.

Your famous!

I have tried a lot of different methods. I tend to try the cheap solder sucker in your first image as a first attempt. After pin removal and sucking as much solder out as possible, I clean up the hole with a hot stainless steel pin. Solder wick is a a last resort for me - it seems to require solder to transfer heat to the braid and it is only hot enough where it is already saturated with solder. Maybe if my wick wasn't so old and oxidized it would work better.

That's the one.  

I have them all and I'm equally inept with all of them.   The one above - the soldering pencil / vacuum plunger - is probably the one I use the most.  I've given up completely on the non-heated solder suckers (probably have 4).  The squeeze bulb type used to be the "Go To" until I discovered the unit above.  I've also got a unit like Andrew_J but I rarely pull it out.  Maybe if I had better space management skills I would have room for it on the workbench.  I do fire it up when I'm harvesting components, but seldom otherwise.  Only recently, I've had success with solder braid.  Maybe I just got lucky this time or maybe it's because I got a new tin of flux and wasn't skimping on using it.

Sidecutters, a regular iron and tweezers to pull the pin out, and wick to clean the hole. 

Thank you for sharing your desoldering experiences! Now I am intrigued I only found out just today these heated suckers even exist, perhaps if I find one cheap enough I will have to take a chance on it.

Yes, if the coponent does not need to be recovered, that's the way to go.

The "fancy" Japanese model is definitely the nicest manual solder sucker I have ever used. The build quality is great. And when it works, it works well. However.

I cannot stand to use it.

The size is relatively small. And I have an exceptionally hard time actuating the button because of the tool's small size and button placement. I end up with this super-awkward handhold so that my finger or thumb can hit the button. So awkward I can NOT usually position it over the joint as needed.

I have watched others use it with no issue. So it is entirely a James problem. But I just cannot figure out how a way to hold and use it. This is a shame, because as I said, the build quality is an order of magnitude better than the cheap plastic ones.

Depending on what I am doing, one option is to use low-temperature Bismuth-based solder. This allows me to remove through-hole components without fully removing all of the solder. Then once the part is out, I can clean it up with a sucker or de-soldering braid.

Thanks for sharing your experience with the "fancy" sucker. 

I think I really should add some of this low-temp solder to my kit. Is there a big difference between the chipquick alloy and bismuth solder?