Is it possible for someone to get into a Ph.D. without a thesis at Masters-level?
If someone doesn't have a thesis at his Masters-level, what other criteria can he fill in order to approach a potential supervisor?
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I assume that you have a Master's degree but not a Master's Thesis. The answer to your question depends on the location.
In the UK and the US, a Master's degree is not a requirement to start PhD studies, and therefore a Master's Thesis is also not a requirement. The admissions committee will compare your achievements with those of other students who only have a Bachelor's degree, and the expectations in terms of research output and thesis writing will correspond to that level.
In other locations, such as Germany, starting doctoral studies without a Master's degree is possible but very rare (unfortunately, the link is in German, but it says that only 1300 out of 108000 doctoral students that started on the winter semester 2011/2012 had no Master's degree).
The traditional way of starting doctoral studies in Germany is to contact a professor directly, and be selected by the professor, without the participation of any admissions committee. Professors will generally expect a Master's degree, and will want to have access to some of your research output to see how good your research is. They will by default look at your Master's Thesis for this, but you might be able to also get selected without a Master's Thesis if you have enough research output (publications or written reports of research projects).
There are also structured doctoral programs in Germany. These function in a similar way to PhD programs in the US/UK (with admissions committee and more course work), and are more likely to accept students without a Master's degree or thesis. These programs are, however, much less common than "traditional" doctoral studies as per the previous paragraph, so you might have a hard time finding one that interests you.
I have advised PhD students who had a masters when they started their doctoral studies, and others who didn't. I did not have a masters when I started my PhD. As noted in another answer, in some countries, having a masters is not required to start doctoral studies. This is, for example, also the case in Hong Kong.
what other criteria can [a prospective candidate] fill in order to approach a potential supervisor?
As an advisor, I would like to see a prospective doctoral student demonstrate ability to work independently. One of the principle differences between undergraduate and doctoral studies will be the expectation to perform independent research; many candidates fresh out of a bachelors degree may have never undertaken more substantial research than writing a term paper. A few examples of independent work could be:
When prospective graduate students without a masters degree approach me about starting doctoral studies, I generally recommend a graduate school path that allows the student to acquire a masters during the course of study. Such an option may provide a graceful "bail-out" option for both the student and advisor, for example if the student discovers after a year or two that research is not for her/him. I realize that such an arrangement is not an option in all countries.
You can still pursue a PhD after doing a non-thesis master's degree. Contrary to popular belief, the thesis master's degree is not the only path to doctoral studies and the world of academia. Although there are a few exceptions, you can enrol in many PhD programs after completing a non-thesis master's degree.
In India, it used to be the case that you can join a PhD with a master degree. It is not necessary to have a masters that requires writing a thesis. E.g., A MCA(master in computer application) (where writing a thesis is NOT part of the course) degree holder can join PhD in computer science. It is not necessary to have a MTech (where writing a thesis is usually part of the course). Now it is also possible to join PhD even with a BTech degree(Again, where writing a thesis is NOT part of the course).