Before getting to the various North American options, just a quick link to a UK discussion
Ceiling fan control - no light (UK)
OK, back to this thread. 
Just to add a visual to @johnconstantelo 's excellent response…
Option 1: Hampton Bay Zigbee Controller. [Discontinued by Home Depot in 2020]. Full integration with SmartThings of both fan and dimmable lights. Optional single gang wall switch.
The Hampton Bay zigbee option has a single gang wallmount Controller that has both fan and light controls and looks like this. This one will also let you reverse the fan if your fan controller has that feature. Because each button is labeled, it’s very intuitive for guests to use.
Option 2: GE Zwave switches. One model controls the fan speed and a different model can dim the lights. Full integration with SmartThings, officially supported.
The GE zwave option will require a dual gang, with one switch to control the lights in the fan and one switch to control the fan speed. They will end up looking something like this. You will get low/medium/high Control for the fan, but no reverse.
http://www.ezzwave.com/z-wave-products/
And here’s a video that shows how the GE switches work when you have one switch for the fan lights and a separate switch for the fan speed. I like this video because the guy walks through exactly how you change the fan speed at the switch. (The app shown towards the end of the video is not the smartthings app, but you would get similar results with the smartthings app, it would just look a little different.)
Option 3: GE Fan Control model 14287 With Doubletap Plus Smart Bulbs in the Fan
GE released a fan control switch in late 2017, model 14287 , which allows you to double tap the top or bottom of the rocker and perform a different function. If you use Zwave (not zigbee) smart bulbs in the fan kit this will work to turn the lights on or off even if your SmartThings hub is offline. 
This gives you a single gang zwave Option, although it will not be intuitive for guests since just tapping the rocker once will only control the fan, not the lights. And you won’t be able to dim the lights from the switch, just on/off. Also, this will only work if your fan kit takes regular full size A19 bulbs. Still, it will be a good choice for some people.
https://products.z-wavealliance.org/products/2506
Although the product description says “lights must be on a separate switch,” that is only true if you want a wired wall switch, in which case use option 2 above. If you use smart bulbs and use the double tap feature of this switch, you can control both fan speed and lights from one device.
https://www.amazon.com/GE-Controls-Required-SmartThings-14287/dp/B06XTKQTTV
https://www.amazon.com/GoControl-Z-Wave-Dimmable-Light-LB60Z-1/dp/B00PJH16UC/
Option 4: Inovelli Red Series Zwave Fan and Light Control
In 2020, Inovelli introduced a single gang minimalist light and fan control. Z wave. Hold down the light button to make the light brighter. Hold down the fan button to make the fan faster. Works with multi speed fans. Includes a speed control device which gets installed in the fan canopy.
Option 5: Lutron Caseta Fan Control Switch*
There has been a smartthings/Lutron Caseta Integration for several years, but they only added the Caseta fan switch to the integration in June 2021.
More details on this later, but if you were already using Lutron Caseta light switches, this will be a good option. Note that like GE and Leviton, the switch only controls the fan speed, you will have to add a second switch next to it to control the lights and use smart bulbs in the light kit.
https://www.amazon.com/Lutron-Wireless-Single-Pole-PD-FSQN-WH-Assistant/dp/B07N1GXM38
All five of these first options work well with SmartThings and you will be able to use Alexa through SmartThings to control them.
Option 6: Treatlife WiFi Fan Control Switch
This interesting device was released in 2020, or at least that’s when it got a manufacturer-provided smartthings Integration through the smartlife app. However, the integration is only for scenes, not for the device itself, which means you have to create the scenes in the smart life or tuya app and then create virtual switches in smartthings to represent those scenes and it all gets pretty complicated. In addition, if someone changes the fan settings outside of smartthings, including by voice or on the switch at the wall, that does not get reported to smartthings.
There are people in the community who have this and do like it a lot, both because of its price and because you can run it without a hub, but the integration just isn’t as good as the first five options.
It is designed to control a dumb fan which is already wired to a dumb wall switch for light control and on/off of the fan, but which uses a pull chain for speed control.
It allows you to set the fan speed on high with the pull chain, then remove the pull chain and operate the fan speed and light from the wall switch. Descriptions I have seen says it is UL listed.
You also need to check the specs to make sure that the fan is not more powerful than the switch is designed for.
As mentioned, Several community members have it and like it. Setting it up with Alexa is a little tricky because it will initially create three devices, one for the fan speed, one for dimmable light, and another one I’m not sure what it’s for. The problem is they all have very similar names. Many people have said they needed to rename the fan speed in the Alexa app to be different from the light in order to get Alexa control of each.
The other issue with this one is that it is a very blocky shape, and some people have said they were not able to fit it into their existing switch box. So check the dimensions before purchase. But it’s a good option to have, and would work in a hub-optional smartthings set up. However, it does require an active Internet connection to work with automations or voice control.
https://www.amazon.com/Ceiling-Control-Treatlife-Assistant-Schedule/dp/B086PPRWL7
And here’s a discussion thread about how to use the virtual switches with it.
Need Help to Integrate Treatlife Ceiling Fan Switch to ST - #5 by Paul_Oliver
Two more retrofit options, but not yet as good as the first six
Option 7: Bond Remote: easy no-wiring solution, but limited features so far and expensive
Olibra has come out with the Bond remote, which is an expensive universal remote designed to add voice control to fans that already have a remote. Note that it doesn’t have its own button remote or wall switch – – it’s really just adding as a second controller to your existing fan system, but a controller that can also understand Alexa or Google Home.
As of June 2019, an official SmartThings integration has been added if you are using the 2018 V3 app.
It costs $99, partly because they have put multiple radios inside this thing so it can work with many different fans that have remotes, on a number of different frequencies or on IR. It doesn’t work with every fan, for example it doesn’t work with a Lutron maestro controller or an Insteon fan controller or some of the Casablanca models. But it works with a lot of the most popular ones as long as the fan already has a remote.
The bond adds voice control and also has Wi-Fi capability and its own IFTTT channel.
It also has a smartthings integration for some models.
It seems to be pretty good at controlling the fan speed, but so far light controls are mostly limited on/off, no dimming. But that also varies by specific model.
One bond is supposed to be able to control up to six fans, but if they have IR remotes they will have to be line of sight to the device. For RF remotes, you’ll just have to try it and see what the range is.
You can contact the company in advance to see if it’s known whether or not it will work with your model.
So the first five options in this post will give you much more granular control of a fan with lights, but the bond is worth looking at if you want a no wiring solution. (Also, the other options in this post all have their own wall switches; the Bond just adds additional integration to what you already have.)
More discussion in the following thread:
@Jamieboy05 , @mamouLA , and @ErnieG all brought up the bond when it was still in pre-release, and I am very glad to see that the device as delivered does have a lot of practical value for some community members. So I have updated this post to include it.
Option 8
As @mwav3 notes below, in 2020 Home Depot released a new less expensive version of the Bond remote Which includes a canopy controller. This one sells for about $30 less than the original. It looks like a handheld remote even if you do mount it on the wall, so some people may prefer some of the more switch-like options.
Option 9: Lutron Maestro. No neutral required, full control, but no status updates from the wall switch
There is one other option if you don’t have a neutral at the switchbox, but it’s not as good as the first four because when someone uses the wall switch the information will not be reported to smartthings so you could never trust the status in the mobile app. So I just mention it here for completeness.
There is a Lutron Maestro fan Control Device (MIR-FQ4FMT-WH) where the Wall Switch can be controlled with a Logitech Harmony Home Hub. It will work fine in terms of having SmartThings control the fan by using the harmony as a “man in the middle”, and you will have a wall switch, but again, if someone uses the wall switch SmartThings won’t know that that happened. So this one is only for retrofit options where the other options don’t work.
https://support.smartthings.com/hc/en-us/articles/205653834-Logitech-Harmony-Home-Hub
If you don’t need fan speed control, just on/off
Option 9) Zooz Dual Switch (light dimmer plus on/off control of fan
In late 2019, the Smartest House introduced a new zwave device which is a single gang size with two separate device controls. The top is a dimmer switch for a light (not for a motor!) and the bottom is a simple on/off button. Early reviews are good. This is particularly popular for bathrooms with a single speed extractor fan. Requires a neutral at the switch.
Summary
Choose one of the first five options if you have a neutral at the switch and you want to fully integrate a multi speed fan and its lights with SmartThings, including keeping the status up-to-date in the SmartThings mobile app. You will be able to use a voice assistant by using the SmartThings integration.
The Home Depot version of the bond may be less expensive, but with slightly less integration.
The treatlife Wi-Fi switch is very inexpensive, but at the time of this writing integration is limited to scenes.
If you are just looking for on/off control of a single speed extractor fan, the Zooz zen30 is a good choice.
If you want a no wiring solution, the original bond is definitely worth looking at. However, at the present time it doesn’t give you full integration.
If you don’t have a neutral at the switchbox but you do want a wired solution with its own wall switch, the Lutron maestro may also be worth considering. You will get control of more features than you might with the bond, but installation is more complex and does require installing a fan controller as well as replacing the switch. And you still have the issue that the status can get out of sync with the SmartThings app.
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