(Almost) any pair of shoes can be swing dance shoes, though you most often see people wearing a mixture of tennis shoes/sneakers/gym shoes, leather soled shoes, and suede soled shoes. Heels are less common, although some people dance in heels (particularly Balboa dancers). Avoid spiky and high heels. Cheap hard plastic shoes often slide better than expensive rubber soles, so if you’re just getting started or are on a budget these kinds of cheap shoes are recommended.
Avoid outdoor running shoes. Especially those with black soles that leave marks on the floor
The most important part is the soles of the shoes.
Slide versus traction. Dancers are looking for shoes that allow them to move on the floor. Some floors are more slippery, and some less, and therefore many dancers have a mix of shoes on the rubber to leather spectrum (sticky/lots of traction to super slippery spectrum).
As a general rule, leather and suede soles will provide more slipperiness, and rubber soled shoes will provide more traction. Cheap plastic shoes often slide better than expensive rubber soles, so sometimes cheapest is best!
Make your regular shoes dance shoes
You can turn any pair of flat soled shoes into dance shoes by gluing leather or suede onto the bottoms. Local cobblers can do this for you, or you can do it yourself. The strongest cement glue is recommended.
In a pinch, you can even put nylons or socks over the front part of your shoe to slide better.
Where to buy
A non-inclusive list of shoe options.
Sustainable and budget options
Use the shoes you already have, shop second hand stores, marketplace etc. Look for hard plastic soles or add suede/leather to the bottoms: How to suede your shoes.
(Somewhat) local shops
HepCat Store https://www.hepcat.se/ (Unfortunately does not sell dance shoes anymore)
Dans på rosor https://dansparosor.se/dansskor/typ/lindy-hop
International swing shoe brands
Slide & Swing https://www.slideandswing.es/en/
Harlem shoes (custom sizing and width) https://harlemshoes.com
Groovy Fox https://www.groovyfox.bg
Saint Savoy https://www.saintsavoy.com/en/
Madame Dynamite https://www.madamedynamite.es/en/
Barefoot brands that work for dancing
Bespokeycraft (somewhat harder sole) https://www.etsy.com/se-en/shop/Bespokycraft?ref=shop-header-name&listing_id=1354419981