How to detach a cleat with a warped screw

If you really want to save the cleat (which looks pretty worn) the simplest option is to simply drill the head of the offending screw out. All this needs is an electric drill and some drill bits. While screw extractors may work, most people don't have them and they wreck the screw anyway. It would take less than a minute to drill the screw out.

Use the existing other screw and select a drill bit that is the same size (or a tiny bit bigger) as the thread part of the screw. Secure the shoe in a vise or something to hold it very steady, put the drill bit on the hex area of the bad screw head, and drill away. The clip will pop right off. The bad screw threads will still remain in the shoe.

Doing this with the other hole empty is hurting your cause. Using another fully lubricated bolt in good condition to relieve preload sometimes makes this situation go easy and is always helpful. You mount an SPD cleat by alternating back and fourth until both bolts are torqued since preload on one relieves it from the other, so you need to employ the same principle in reverse here. Grease the shoulder and tighten it to the brink of breaking something.

The bolt head doesn't look so destroyed that a good 3mm wrench in good condition will get nothing on it. If that is the spot you're in, using a dremel cut-off wheel to make flathead screwdriver flats usually works well on SPD bolts. And then if that doesn't work, just go the rest of the way down with the mini end mill dremel bits to obliterate the center of the bolt head.

Some shoes are put together in a way where the tongue and insole can be made sufficiently out of the way to use the plate as leverage from the inside. In other words, if the plate is resting on the sole as opposed to being inset, you may be able to twist it with the other bolt out. If you were up against corrosion as opposed to preload, that approach could also work.

A dremel/rotary tool made this really easy last time it occurred to me. Just cut a slot, load a flat head bit into a ratchet, and zip it right out. Cut the slot nice and deep.

I have extracted warped/partially-stripped hex head screws many times by using flat screwdrivers. Usually I use a T-handle that accepts modular screwdriver bits, and I go through my collection of different-size flat screwdriver bits until I find one which I can jam into a pair of opposing corners of the roughly-hex cavity. I often have to lean the screwdriver over to make the screwdriver bit dig into the screw I'm trying to remove, so as not to spit my screwdriver out when I apply torque.

Also, some penetrating lubricant like WD-40 or Liquid Wrench (which is best kept out of the cavity where you'll be trying to jam in the screwdriver, as it's slippery) can make the job easier, by loosening the rust I would assume is helping bind the screw to the mounting plate.

I don't see getting that screw out without ruining it, so what is the point? It's unlikely that you have another screw that can successfully replace the ruined screw. You have new shoes, so get new cleats.

When you attach the new cleats to your new shoes, put some grease on the screw threads before installing them. That should make it a lot easier to get the screws out someday, without ruining the screws. This is a step that a lot of people neglect to do, and that's probably why you are in the situation you are in right now.

Since the cleat is stuck on your old shoe, you can destroy as much as needed of that old shoe to get at the bolt with a hacksaw.

Always be mindful of what parts you want to save, and what can be destroyed. If you're throwing away the shoe, it becomes expandable.

But let's be real, buy new cleats.