- Officially Compatible
There’s only one garage door controller on the official “works with smartthings“ list, and that’s only for the US. It is UL-certified and very popular. That’s The Linear/GoControl/Iris/ADT/NuTone zwave model ( all made by Linear, just rebranded)
- Telguard Zwave
Telguard used to make a simple Z wave one, but they were acquired by Ametek and then seem to only be available as part of amatek home security package . Some community members are using them, but they’re hard to find now. If you do get one, it just appears to smartthings to be an on/off switch.
https://www.telguard.com/product/gdc1/
- Aeotec zwave
And Aeotec got one zwave-certified for both the US and EU, but as they have sometimes done with other devices, they appear to have released a single production run and then let it go out of stock everywhere. It’s unclear whether it will ever be restocked or not.
(As with all Z wave devices, remember that the frequency of the device must exactly match the frequency of your hub. A US hub will only work with US frequency devices, a UK hub will only work with UK frequency devices.)
https://uedata.amazon.com/Garage-Door-Controller-ZW062--Aeotec/dp/B0151Z9WUM
You do occasionally see The European version show up from third-party sellers on some of the Amazon European sites:
- MyQ
There’s an unofficial cloud To cloud community integration with MyQ. This is also very popular.
[RELEASE] MyQ Lite Door and Lamp Control (for Liftmaster/Chamberlain)
- Other WiFi Options
people, particularly in the UK but also in the US, often ask about other options.
In September 2019, CNET ran an article listing a half dozen Wi-Fi garage door controllers. Most of these work with IFTTT and you can get some SmartThings integration that way.
These include garadget, gogogate2, Garageio, and a few more. Some are available in the U.K.
Wirecutter Also has an in-depth review article of some of the top models, including MyQ, the Linear/GoControl, and Gogogate2.
- The non Safety Certified Option
Yes, you can essentially hotwire your existing garage door controller by just putting a cut out relay on it and turning the power on and off. But it’s just not a good idea for something you might be using when you are out of sight of the garage. Garage doors can kill people or pets. Now that the cost of the safety-certified versions have come down significantly, It’s worth the extra $30 or $40 to get a system that has the required remote control safety features. 
Custom DTHs
There is community developed code for garadget:
Garageio:
Telguard:
[Release] Telguard GDC1 Device Type
For other brands, see the quick browse lists in the community – created wiki and look for the “garage“ lists in the DTH and Project Reports sections: