Here's a giant cool story, but i've kept bettas now in pretty much every way.
It really is true when people say that it is dependent on the betta, with both males and females. It's a good idea to put your tank in a place where you can glance over often and check on how they're doing with the fish.
Generally if they are nippy, or chasing other fish around on a regular basis then it isn't a good fit for either. Bettas enjoy patrolling community tanks just as they do their own, but if they do it too much they will become exhausted and will be weakened to community diseases.
It helps if you stock around your betta, try to layer your tank with the idea that your betta is the king of the castle.
Give him an area at the top of the tank that's sheltered, floating plants are nice here because they do enjoy beaching themselves, just secure your lid, as well as an area at the bottom of the tank to hide. I like making a plant/rock line 3/4th back into the tank, which makes a really good patrol easy/safe/stress free area for your betta. Avoid sharp decorations and NO plastic plants. None. Absolutely not one. Silk or real only.
There are plenty of low light plants that do fine with very little, and really do help the water quality if you maintenance them at least once a week. Also low light aquatic plants are a cinch compared to terrestrial.
As far as possible tank mates go with willing bettas,
1)smaller cory species that do not seem to bother bettas, I've even had luck with them and more aggressive specimens.
2)African dwarf frogs make good tank mates, but finding healthy stock is really hard, so before you pick them up make sure to look up their common illnesses because they are rampant in these little guys.
No african clawed frogs ofcourse.
3) Cherry barbs have done surprisingly well with my bettas, in a 15 G, you could keep one male with a harem of 3-4 girls which is actually surprisingly colorful. Males deepen to a dark red, which do not seem to cause aggression from any bettas I've kept them with, and girls are orange and gold striped with dark brown and the cutest lip stick!
4) Live bearers are a toss up with bettas, I think they make tanks too messy and trash water quality, just be careful with guppies as they do tend to be aggressive towards them, though I've had a few bettas that really couldn't care less.
5) Neons- Black neons tend to have less issues than the red and blues.
6) Danios- Work IF IF IF You give them a proper school. If not there will be either one, or a few odd men out that will pick at the betta and make him very grouchy.
7) Hatchets- if you keep a tight lid on your tank, and invest in a spray bar to agitate only the surface current it can work, but I find that it keeps bettas hanging towards the bottom of the tank, which they really don't seem to enjoy.
8) Smaller loaches- actually make a pretty amusing tank, the key here is hiding places and water quality. with a 15 gallon I would hesitate, but it might be possible to put 4 queen loaches in there, with just a betta and plenty of hiding. I'd look for adult specimens that are on the small side. Research into the loach type carefully, as they vary from 2-3 inches, to 1-2 feet.
9) You could do an all female tank, with no male betta at all as a 15 G is way too small to attempt a harem tank. They look really dull in the store, but they really do brighten up beautifully, especially together. The key here is to add them all at one time, and monitor carefully as they establish a pecking order.
10) You can also either purchase, or make a divider by cutting up one for a larger tank, and house two male bettas. With this set up I always suggest two small internal/hob filters on either side with sponge over the intakes, and any gaps that might lead inside the filter. If you have an issue with current still, I just pop bits of sponge into the filter behind/infront of the media, works just fine. Hot glue guns are also handy for this if you want it to look clean, and i've never had an issue with using it in the aquarium.
Bad tank mates that seem good
1)Ghost shrimp are betta food, have never had luck with keeping the two together.
If you had a larger tank bamboo shrimp would be an option but it will most likely starve in smaller tanks, I've had no luck keeping them red in tanks under 30 gallons personally.
2) Ghost catfish- need too high of a current to ever live happily with a betta
3) Molly- Really should be kept as brackish, messy messy messy fish!
4) Otos- Really need more than a 15g to have adequate food without it being a pain, also a higher current, and are best kept in schools.
5) Any gourami- Both fish tend to be very aggressive towards each other, and if you house two together peacefully, its because they both are generally peaceful exceptions-- not something to ever bank on, and with these combinations it is rarely fruitful.
I could go on and on about the combinations, so if you have any questions unaddressed, I might be able to help you.