I really liked the Logitech F510 Rumble Pad but the cord completely fell apart in only a year. I'm just wondering if any of you guys have played with a good gamepad in the $30-35 range that is a bit more durable.

I would much prefer corded, I don't like having to deal with receivers and batteries.

Wired X360 controller is my preference.

An Xbox 360 pad should be fine?

I bought a used XBOX 360 controller and love it.

Certainly Xbox 360 controller.

Depends on what you want to play. The Xbox 360 controller has a nice build quality & the sticks are quite good on it, but the D-pad is garbage, so if you want to play 2D platformers and the like, don't get a 360 controller.
I'm personally using a Logitech Cordless RumblePad 2 and I'm extremely happy with it. It has a good weight to it, it feels accurate and the shoulder buttons are among the best I've ever seen on a gamepad (considerably better than the 360 controller). The sticks are not quite as good as that on a 360 controller though, so if the games that you intend to play relies heavily on those, then a RumblePad 2 should not be your first choice. The D-pad on the other hand is really good.
There exists a corded version of this controller as well.

Maybe later, right now I want something with a good dpad.
I might just get another F510 and grab a 360 pad next week.

Seems I get only get the rumblepad 2 refurbished for a decent price,

You could use a PS3 pad with MotionJoy. I also use a PS2 pad with a USB adapter as well as a few old Saitek 6 button pads for fighting games and a few other things. I use a wired 360 pad for certain things made for 360 pads like Super Meat Boy or Dark Souls. The dpad is garbage though so keep that in mind.

I've had both a wired 360 pad and a Logitech F310 (which is just a 510 without a vibration motor) and I would peg the dpad's about even. The Logitech allows for depression of the dpad at the center of the cross to allot an extra button, which causes some goofy depth issues and can lessen effectiveness in games that require sweeping movements (like fighting games). The X360 dpad has difficulty with providing tactile feedback on movement direction and sports an odd surface texture, but (in my opinion) becomes serviceable after some extensive practice. I wouldn't call it garbage, I'd just say that it's unconventional, and not in the way the Logitech's depression feature is unconventional. I could never quite get over the "extra" button embedded within the Logitech and found myself defaulting to the 360 pad. The Microsoft pad's construction quality and superior face button and trigger placement seal the deal for me crowning it a better pad for almost any game type. That's just my two cents.

Why buy a 360 controller? There are far superior alternatives out there. Personally, I use a PS2 controller- you can pick up a USB converter real cheap off eBay. Alternatively, a GameCube controller may also work, but the placement of the A button makes any game which requires you to constantly move between three face buttons incredibly difficult to play.

While it looks really uncomfortable, I love the idea of a switchable D-Pad and analog stick.

You can get a genuine DualShock 3 and use it with MotioninJoy on your pc. It is a joy to play games designed for a gamepad with this controller.

You can use it in wired mode by connecting it to your computer with a standard usb cable. That way if something goes wrong with the cable, you will only have to replace the cable, and these cables aren't expensive.

Since so many praise the XBox360 controller, it doesn't have issues with GOG games like the Rayman games due to missing DirectInput support?

I use a PS3 controller and MotionInJoy, but thats only because I already had a PS3 and the software is free. However, it needs to emulate the XBox controller to work properly in some games. Not a problem though. By the way, I use the wired connection mostly, but the built in bluetooth works nicely also.