Reebok Zigs and Nike Shox
If it were up to me I would outlaw these shoes and draw and quarter anyone who had a hand in their creation. These are by far the worst shoes that you could wear to lift weights and/or live your daily life in. The heel-centricity of these shoes basically means you'll live the rest of your life with tight ankles, hamstrings and hips and be weak as a kitten with MD in the same areas. I won't bore you with the biomechanics of the foot, ankle, knee, and hip but suffice it to say these shoes blow.
I can't hear you. Your shoes are too loud.
Every kid I've seen wear these shoes has horrible ankle mobility, weak glutes, and tight-as-piano-string hip flexors. This stems from the fact that you are basically putting on four inch pumps when you slide these guys on. Talk to any women who wears heels on a regular basis and she'll tell you. It ain't fun. The raised heel put a lot of stress through the hip flexors. This causes them to become tight and pull the pelvis into an anterior (forward) tilt. This anterior tilt tightens and weakens the glutes and hamstrings rendering you all but useless when it comes to any type of pursuit that involves any kind of physicality.
The raised heel also shortens the calves which will adapt and stay that way over time. Tight ankles mean that the knee can't bend over the foot when doing anything with the lower body. If you want a recipe for disaster in the form of lower body injuries this shit is the bomb. The Achilles tendon and ACL can only take so much.
What To Wear When Smashing Weight
Every person who has spent a good amount of time training and is above average in strength has a personal favorite shoe to wear. Some like Chuck Taylor's, some like indoor soccer shoes, and some, like me, prefer wrestling shoes. If you can picture all of these shoes they all have something in common: a relatively flat sole that rides close to the ground. Having a flat sole with little cushioning allows you to A) Create space at your hips while squatting, dead lifting, and pressing by being able to sit your hips back. This is what engages your glutes and hamstrings along with your quads. The flat sole also allows you to push your weight to the outside of your foot, keeping your knees towards your pinky toe. If you like your ACLs then you'll try to keep your knee to the pinky toe.
An oldie but goodie.
One caveat to this is that some weirdos prefer to wear Olympic weightlifting shoes. These are shoes that have a wooden, raised heel that is about a half to a quarter inch up. Doing the Olympic lifts in these shoes definitely help, but for the majority of gym lifts I could take them or leave them. For some people the biomechanics of their own body makes these shoes a better, ahem, fit. I've tried these shoes for squatting and other movements in the past and have had mixed results. I like to wear them when I'm trying to target my quads a little more, but don't wear them while doing max effort power lifts. The difference in these shoes and the Zigs and Shox mentioned above is that the sole is completely solid. It's very hard to fall back on them because the heel doesn't give. It's hard and made of wood. Also, they are ONLY worn during lifting. The same things would happen to my lower extremities if I wore these all the time. So don't do it.
My Two Cents On Barefoot Running
Barefoot running or running with minimalist shoes like Vibram's has been gaining a lot of traction as of late. One of the reasons for this are studies that watch African and South American tribesmen, who never wear shoes, run. They strike the ground with their toe or mid foot first with the heel never really making any significant contact. The studies then go on to say that they have healthy joints and muscles and blah blah blah. I don't disagree with the studies. In fact, I fully agree with them. The one detail that gets lost on a lot of people is that THEY HAVE NEVER WORN SHOES. These tribesmen didn't wear heel-centric shoes their whole lives, read an article in a magazine, and suddenly decide to run barefoot. They've been doing it their whole fucking lives. As such they have developed the movement pattern and the support structures (bones, ligaments, tendons, and muscles) to support running barefoot. If you are going to give barefoot running a try you first need to give running with shoes with less of a heel drop a try. Then try flat soled shoes. Then try barefoot. You need to build up your body to be able to handle it.
But that's only half the equation. If you've worn shoes your whole life then you have a movement pattern ingrained in you that requires shoes. It takes 10,000 perfect repetitions at something to make it a habit. To make the switch you have to unlearn how to run with shoes and relearn how to run without them. That takes a lot of time. Like months to years of knowing what the hell you are doing. And in the instant gratification society we live in not a lot of people are willing to do that. Or even realize they need to do that if they don't want to hobble around the rest of their lives because they were too lazy to do something the right way. I better stop before I completely change directions. Remember at the beginning when I said I had no authority to tell you what to wear when running? Yeah, I lied.