About using static classes:

Maybe it is a matter of taste, but for my understanding static methods are functions that depend on input parameters only. If you have a single object that contains state, it is better to use a singleton or even better to use an DI framework.

So, the main disadvantage are IMHO that other developers may not expect a state behind a static class and it is not possible to mock that class (e.g. for unit testing). See also this discussion.

Code review

• The field DeviceConnected is not set from outside so it is better to create a property instead: DeviceConnected { get; private set; }.
• The class ComPort is not threadsafe but it is used from different threads. I can not see all usages, but that is generally not a good idea (even if it works, maybe somebody else has to change it some day)
• The methods ComPort.Read, ComPort.Write and ComPort.Close do not check if the device port is connected and will probably throw an meaningless exception. Maybe it makes sense to handle that or throw a descriptive exception.
• Not sure if that is desired, but if the device does not exist, the loop while (!DeviceConnected) { /* .. */ } will run endlessly. Consider to use a timer instead of the while loop.
• If method SearchCOM takes longer, it is better to start the task in view model with option TaskCreationOptions.LongRunning. see also

About using static classes:

Maybe it is a matter of taste, but for my understanding static methods are functions that depend on input parameters only. If you have a single object that contains state, it is better to use a singleton or even better to use an DI framework.

So, the main disadvantage are IMHO that other developers may not expect a state behind a static class and it is not possible to mock that class (e.g. for unit testing). See also this discussion.

Code review

• The field DeviceConnected is not set from outside so it is better to create a property instead: DeviceConnected { get; private set; }.
• The class ComPort is not threadsafe but it is used from different threads. I can not see all usages, but that is generally not a good idea (even if it works, maybe somebody else has to change it some day)
• The methods ComPort.Read, ComPort.Write and ComPort.Close do not check if the device port is connected and will probably throw an meaningless exception. Maybe it makes sense to handle that or throw a descriptive exception.
• Not sure if that is desired, but if the device does not exist, the loop while (!DeviceConnected) { /* .. */ } will run endlessly. Consider to use a timer instead of the while loop.
• If method SearchCOM takes longer, it is better to start the task in view model with option TaskCreationOptions.LongRunning. see also