United States
TheCubicle
Sponsors World Class Speedcubers, such as Max Park and Tymon Kolasinski. Based in New York state
SpeedCubeShop
Sponsors other world class such as Antonie Paterakis
Cube Depot
Based in New York City.
iCubeMart
Based in Los Angeles.
Canada
CANcube
Based in Ontario, Canada.
Mexico
Australia
New Zealand
Speedcube.co.nz
NZ counterpart of speedcube.com.au
United Kingdom
Denmark
France
Variantes
Also has a physical store in Paris, 29 rue Saint-André-des-Arts
Germany
Netherlands
Spain
Sweden
Finland
Poland
Romania
Switzerland
Argentina
Brazil
Bolivia
South Africa
China
Chinese shops have the best prices and some of them ship worldwide for free.
The only catch is that shipping times to North America and Europe are generally in the order of weeks to one month, sometimes more. If you can't wait for a month, don't buy from those sites.
Specialized stores
CubeZZ (free worldwide shipping)
Very competitive prices store-wide. They often have sales on popular new speedcubes. However, their range is not as wide as HK Now Store's when it comes to exotic puzzles. Shipping is free, a tracking number is $3 when you ask for registered shipping. Their free shipping normally takes 3 weeks, but you can be unlucky sometimes and have to wait a month or longer. They also have great customer service, and will reply quickly to questions or issues, including shipments that are overdue. They stock large cubes such as the Moyu 5x5 and 7x7, but for legal reasons they don't show them on their website. To buy those, contact them directly at service@cubezz.com. The list of larger cubes and their prices as of February 2020 can be seen here.
ZiiCube/ZCube
Insane value for money, especially for bulk orders. One should be careful if they are ordering a large number of cubes (>100) and they choose DHL Express, as they may be charged large amounts in customs. You can get VIP membership after spending 300 RMB (around $45) which reduces prices to what they would be if you ordered 2 cubes instead of one.
The only drawback is no free shipping, but shipping is a lot faster than the free shipping options of other Asian shops. Due to corona there are discounts on all products, so overall it's still cheaper than most other stores. For very small/cheap orders a shop with a free shipping option could be cheaper.
HK Now Store (free worldwide shipping)
Home of "Calvin's puzzles". Lots of choice, both in speedcubes and more exotic puzzles. Good customer service. Shipping is free, a tracking number is $2.
ChampionCubeStore previously known as 51morefun
Other chinese stores
Wallbuys (free worldwide shipping)
Fasttech (free worldwide shipping)
Icebay (free worldwide shipping)
India
https://thecubestore.in/ (formerly pcubedpuzzles)
Indonesia
Japan
Kazakhstan
Malaysia
Russia
Singapore
South Korea
Thailand
UAE