Are appeals of termination from PhD programs common and are there any unambiguously successful or unsuccessful approaches to fighting a termination? I'm currently in the process of appealing a termination from my PhD program and although there are many facts concerning my case that I feel are relevant and important, I'm not sure how relevant they are in the consideration of my appeal. 

My question is: how do I present my appeal in a way that ensures I have the best chance of success? 

This may seem subjective, but there may be some objectively unsuccessful approaches to appealing such a decision. Are there any objectively successful ones? 

The main thrust of my department's decision to terminate my candidacy is that I lack basic competency in topics related to my research. The issue is that the way my competency was evaluated was not in line with university and department guidelines (generally we are evaluated based on reports and formal meetings), I was evaluated based on my response to an email requesting more information from me about my research. In the grand scheme of things, this is a small part of the processes leading up to my termination, but it's what initiated the process. Would it be reasonable to appeal such a decision? 

It is true that I don't have competency in the topics I was evaluated on, but these were part of one project I was working on and not related to the main thrust of my research, so I'm afraid I'm almost like a "fish" that was evaluated on its "climbing" ability (if that makes sense). I am confident I can "swim" quite well (meaning, I have other research merit as a PhD student), but there is no way I could "climb" in the time I was given.

It's simply nerve-racking and difficult consider my own assessment of myself over the assessment of experienced professors that seem to be making objective decisions about research ability, so I'm not sure whether my concerns are valid or not.

FYI - this is in Canada.
Typically, successful appeals happen when another advisor is willing to take on the student after the appeal. This creates an ally during the appeal process. It also creates an "out" for the department. I have two friends who were able to appeal by finding an alternative advisor. I know no-one who was able to appeal with their original advisor UNLESS the advisor was actually using the process as a "wake-up call" rather than an authentic attempt to force the student out.

In the cases where an advisor is using the process as a real authentic attempt to force a student out, the advisor is communicating that they no longer desire to have you as a PhD student. The department is going to be reluctant to force that advisor to continue with you, despite the soundness of your logic. The reluctance is partially because it creates an uncomfortable/unproductive mentoring situation when one party is not enthusiastic. The reluctance is also related to an overall ethos of not wanting to oppose faculty on issues of academic judgement.

The presence of another ally/advisor who will take you on as a student after the process is concluded allows the department to evaluate your claims independent of their reluctance to force your original advisor to take you back. It allows everyone to put this down to a matter of "fit" rather than genuine problems with your work or your advisor's judgement.

IF your advisor is actually on the fence about this, you could attempt to convince your advisor that you now are "awake" to the stakes and they will find working with you to be much better/more productive in the future.