Whether/how to list a failed student on my CV I'm a postdoctoral researcher.  My professor recently asked me to co-supervise one of her doctoral students, whom she also employed full-time as a research scientist.  Since I had no prior experience advising at the doctoral level, I gladly accepted.  Unfortunately, the guy turned out to be a phenomenally poor researcher.  He was habitually late for work, took absences without leave, lacked subject-matter knowledge that we expect even of undergraduate students, and had extremely poor oral and written communication skills.  I worked my best to help him, and we even managed to get a paper published (though I had to completely rewrite his draft, many parts of which were completely nonsensical or even plagiarised).  However, his bad work habits continued, and despite my tutelage he remained unwilling or unable to fill the gaps in his knowledge.  After about six months of my co-supervision, my professor decided to dismiss him from the doctoral programme and from employment.

I'm currently seeking employment, including tenure-track positions.  My question is whether and how I should list this failed student on my CV.  On the one hand, I think it would be helpful to show that I have at least some experience advising at the doctoral level.  But I don't want it to reflect poorly on me that the only such experience I had was ultimately a failure.  Should I list him or not?  And if so, what explanation, if any, should I include in the CV as to why the supervision didn't continue past six months?
One thing to keep in mind is that tenure-track faculty generally do not list failed PhD students on their CVs.  One might argue that they should actually, since -- unlike a non-academic resume -- a CV is supposed to be a factual, comprehensive record of one's academic career, but nevertheless academic culture doesn't cut this way.

The next thing I will say is that in my experience at least, "co-advisor" refers to a member of a group of more than one faculty member -- usually, though not absolutely always, tenure-track faculty members -- who jointly share the role of advising.  You seem to be using the term as a kind of "lieutenant advisor," but that could be misunderstood.  When the student's other advisor is your PI, it is likely that your advising duties are less official and more subsidiary.  In particular, you write "[M]y professor decided to dismiss him from the doctoral programme and from employment," not we decided...Nor did you hire the student, fund him and so forth.  However, from the sound of it, you played a large intellectual role in supervising the student.

So: I would not recommend you put a line on your cv which says "co-supervised a PhD student."  If potential employers are interested in that, they are likely to ask you about it, and when they find out that the student didn't graduate, I think that sounds bad...in particular, it sounds not completely forthcoming.  On the other hand, "co-supervised a PhD student who did not finish [or however you want to put it]" also sounds bad, as I mentioned above.

I would suggest instead that you put a line on your CV saying that you mentored a PhD student and that mentoring relationship resulted in a joint paper.  That's a successful mentoring relationship.  The student was not ultimately successful in the PhD program, but I claim that you're not on the hook for that: you didn't hire him and you didn't fire him.  If they ask for more information about the PhD student, you should of course be completely forthcoming about the fact that he was dismissed after six months, as well as all the help you gave him.  If/when this additional information comes to light, it does not undermine what you said before in any way.  Since you are not claiming the PhD student as your own, his fate is not your responsibility (which is true!), and so your involvement in the situation looks (and was, based on your description) entirely positive.