I'm new to clipless shoe and I've seen some non-exposed cleat (MTB type), but is there such a thing as a for clipless shoe that is "flexible enough", which can be used for running or indoor gym like basketball or soccer?

If such a thing exist which would you recommend? Or is it most MTB type shoe would work?

Secondly, what the purpose of overshoe when you can get a full length lycra trousers

Thx

If such a thing existed, I'd be worried about the cleat contacting the floor, and would be a bad compromise for both uses. Too flexible for cycling, too rigid for running.

Overshoes ("booties" or shoe covers) are for keeping your feet warm in cold weather. The heavier ones are made of neoprene and allow you to ride in sub-freezing temps.

I doubt that such a thing exists. You are looking at different purposes and different needs. A shoe that you run in needs some impact resistance and some flexibility for your foot. A bike shoe doesn't need impact resistance nor flexibility. Flexibility is actually a detriment to bicycling since you are pushing down on the sole of the shoe to propel the bike forward. If the sole flexes downward...along with your foot...and you end up with painful pressure points.

As for using any kind of cleat even near a basketball court, I'd say do it only if you really want to end up in the basket

Shoe covers, if that's what you are talking about, are usually used in cool or cold weather. Bike shoes are generally made for summer use and are well ventilated. That's not good for fall and winter temperatures. I don't know of any lycra tights (the proper term) that have integrated shoe covers nor would they be all that easy to use. Pulling tights on...or off...over shoes isn't all that easy anyway. If you have to also pull them over shoes and get the holes for the cleats aligned, it'd be a nightmare.

Sort of off topic, but what sort of shoes do cyclocross people use? There's a fair bit of running around/over obstacles in that, right? Probably for the OP, a backpack/pannier with gym shoes in the best option, but I've always been curious about the xcrossers.

Most wear mtn bike shoes and use pedals that are easy to clip into, like Eggbeater or Candy.

I agree with cycocommute. I have a pair of 661 SPD sandals and they give me a hot foot due to too much flexibility in the SPD cleat mounting area. I find my Shimano SPD sandals and Specialized SPD shoes to have the best compromise between walking flexibility and SPD cleat required stiffness. The Specialized shoes have a all rubber appearing outer sole and the label says Taho MTB. An excellent compromise between walking and riding but I would not want to use them as basketball or running shoes.

Did you get clipless pedals with enough of a platform to allow pedaling in non-cleated shoes? If so I would just ride in regular shoes if you're planning on playing gym or field sports later. If your pedals don't take regular shoes very well, you can always through some gym or field shoes into a pannier, messenger bag, saddlebag, or basket. You can also make arrangements to leave your shoes at the gym in a locker, or have a friend bring them with their gear.

As others have said, shoe covers help add another layer or warmth and dryness. Whether you need them or not depends on your shoes, socks, pants, whether you use fenders, the weather, etc. I've only ever used them with racing shoes since that's all most of them will fit... and racing shoes are what need them anyway since they are usually made for keeping feet cool.

My commuter bike has pedals that are platform on one side and SPD compatible on the other. I often use my commuter bike for long rides over varied terrain and like to wear my mountain bike shoes and clip in to the SPD side. Other days I just wear whatever shoes I want and pedal on the platform side. It's nice because I can wear warmer socks and shoes in the Winter.

If I did cross, I'd use my Sidis. I got the Giau model at a good price:
https://www.sidiusa.com/sidi/mountain..._mountain.html

I used to use Specialized Taho shoes. They were just comfy enough to spend the day in them. No way I would ever use them on a gym floor, though.