When should a student attempt unsupervised research? I just finished my masters and am currently working as a research assistant before starting a PhD in applied mathematics. I was doing some preliminary work for a possible project and derived some minor results which I find interesting, but in the end, we decided not to go ahead with the project.

I still find the minor results interesting and thought of digging a little bit deeper by myself. However, since the group is no longer interested in this, I would have to do this by myself, unsupervised. Is this a very bad thing? Or a good learning experience?

I had a bad experience doing unsupervised research (wasted a lot of time for no results). However, I know that as a PhD student, I must eventually publish first-authored papers. How do I know when to make the switch?
It is a common thread - see &quot;Go rogue too soon/too late&quot; in 10 easy ways to fail a Ph.D. and Going Rogue at The Chronicle of Higher Education. An excerpt from the first:

The advisor-advisee dynamic needs to shift over the course of a degree.
Early on, the advisor should be hands on, doling out specific topics and helping to craft early papers.
Toward the end, the student should know more than the advisor about her topic. Once the inversion happens, she needs to &quot;go rogue&quot; and start choosing the topics to investigate and initiating the paper write-ups. She needs to do so even if her advisor is insisting she do something else.
The trick is getting the timing right.
Going rogue before the student knows how to choose good topics and write well will end in wasted paper submissions and a grumpy advisor.
On the other hand, continuing to act only when ordered to act past a certain point will strain an advisor that expects to start seeing a &quot;return&quot; on an investment of time and hard-won grant money.
Advisors expect near-terminal Ph.D. students to be proto-professors with intimate knowledge of the challenges in their field. They should be capable of selecting and attacking research problems of appropriate size and scope.

Personally, I have a lot of experience with unsupervised research (yes, usually wasted a lot of time for no results), starting from high school.
Sure, doing advisor's project is much safer (as, in some sense, (s)he is responsible for the big picture, references, judging which result is good enough) and allows to forget about doubts (for good or bad) and concentrate on work.
However, what I've discovered after 1.5 of my PhD (mostly &quot;going rogue&quot;) is that there are two important things:

asking experts (or peers), and
day-to-day collaboration.

Sure, you can have an expert and a collaborator in the same person of your advisor, but as long as you are not afraid of talking to other people and have some knowledge in the topic, it shouldn't be hard to ask experts appropriate questions.
When it comes to collaboration - it may be trickier (but not if you locally have people with an interest overlap).
On contrary, working alone for months on a theoretical problem, with little feedback, may be very hard (both technically and psychologically). I've learnt it the hard way. It is not a problem for one research line, but it is when all research is done alone (may be inefficient and depressive, perhaps unless you are in a deep love with it).