I am at the point that I have owned nearly every fish, invertebrate, or frog that most people list as being compatible with bettas. I've found a lot don't work for very good reasons. So let's talk about them!
White Cloud Mountain Minnows
I've recommended these to people before. I owned a school for a long time, but I recently re-homed mine. They're great in terms of how most bettas react to them. Due to their dull colors, they tend to not trigger aggressive bettas. They're beautiful and slow moving fish who won't stress your betta out by too much activity.
However, don't get them for your betta tank. I quickly discovered after owning them that one would die off every 2-3 months. While it would normally not be an issue, these fish live up to 15 years. Which means their life was cut pretty short. The reason?
It took me a while to figure out what their symptoms were. Disease? Stress? What was going on? Turns out: It's far too hot for them. A lot of sites list them as being able to survive in the temperature range bettas are in, but more reputable sites say this range is far too hot and will drive them to an early grave.
A lot of people recommend these for bettas because they are bottom dwellers. What they don't tank into consideration is that most loaches are nippy little jerks. It took me a while to figure out what was going on with Tony's tail tearing up (You may have noticed how tattered it always looks. Sadly it doesn't seem like he will ever heal from this.). I removed decorations, blamed it on and treated for fin rot... everything.
Then one night I saw one of my loaches latch onto his tail as he freaked out and it tore a big chunk out. I re-homed them the next day and removed Tony for that night. After reading up on it, I discovered that they are VERY nippy with long-finned fish. A plakat may be alright with these guys, but veiltails, halfmoons, crowntails, or double tails should not be placed with them.
Again, these are the nippiest suckers I've ever seen. I never placed a betta with them, but when I was younger we had some briefly with a longer-finned species of fish (I can't recall what it was now). It was a larger school, I believe, and they tore the other fish to bits. Some people say not all tetras are quite so nippy, but I've heard such horror stories and seen what they can do first hand that I would never trust them with my betta.
Their colors, speed, and activity can also stress bettas out. Especially neon tetras. Just because your betta can't attack them because they're too fast does not mean they are compatible.
Never placed these with bettas personally, but I have a lot of secondhand experience on the subject.
I've never seen these on any list except to list them as "DO NOT PUT THESE WITH BETTAS". However, I recently asked a prominent betta community for stocking advice and I got advice ranging from tetras (after I had listed them as something I do not want) to guppies to platys. I was a bit boggled by this. Platys are extremely colorful livebearers that bettas just can't seem to stand.
They also can become aggressive and nip, on top of preferring different water parameters than bettas.
There are, of course, others like guppies, cichlids, and goldfish, but... I thought I would do ones that tend to be recommended.
Note: This is largely based on personal or secondhand experience. There are always exceptions to behavior in certain types of animals, so those listed as aggressive or nippy may not be towards your betta. I still would ask you to exercise reasonable caution when introducing your betta to those fish.