What social skills should I practice to make a successful doctoral application? I've been applying to doctoral positions in mathematics with a strong background. BSc and MSc in math, and a MSc in computational math/engineering. I've got research experience in my field and connected fields, a paper in preprint and more to come (with novel results), solid references, teaching experience, participated in international competitions representing my university/country, good grades. Still, most of the time I don't even get an interview. I definitely understand that there are people who are better than me, and I'm not mad about it. However, often when they publish who got the positions, and it can be someone who did their BSc over 6-7 years, or who had an adjacent degree (such as computer science or physics) but studied more mathematics, or who hasn't done any teaching or research otherwise.

A few professors I know have told me several times that it's not really fair, so what I have left are social skills. I just don't know what to say. I've had my CV, statement of purpose, application letters reviewed and they said it was really good.

How can I write an application that broadly speaking appeals to such a recruitment committee? What are some interesting forms of describing my ambitions? What catches the attention of a recruiter in terms of research ambitions? I have clearly stated my research topics, as well as why I want to do research in it and how, but I don't know how to spice it up.

Just to be clear, I'm in Europe, so I didn't drop out of any PhD program to get my master degrees. Also, most of the positions were open positions, so I just sent my application and then they will decide on what advisors should get new students.

EDIT 1: I'd like to clear out that I have a BSc and MSc in mathematics, and an additional MSc which is much more applied and incorporates a lot of physics and computer science. Thus, I do have the required courses to apply for a pure math position, which I even have checked up on on their websites.
It sounds more like an issue of relevance and application strategy rather than a lack of your part, such as social skills.

To answer the question as is, showing that you can balance independent/ individual work and being part of a group of people - not necessarily a research group but that you can network on your own, seek and offer advice and take initiative in communication - is a good approach. With the experience you mention, I believe you can find tangible examples on both aspects. 

However, not getting interviews is a different issue. Both applying for a PhD and selecting a PhD student has a significant personal element. At the very least, both you and the potential supervisors must choose whether you want to work with each other for at least 3 years and produce something never done before. The supervisors may rely on past experience to make a decision. Also, it is not necessarily the objectively best candidate ("strongest") who will be accepted, but the fittest (with or without quotes) given the approach and culture of the department and the academics. Your background can be both an asset and a hindrance. A mixed or checkered background may be a better match in multidisciplinary or far-fetching projects, and may also reveal something about the person (e.g. adaptability, perserverence, wider experience etc). A "pure X" background provides obvious advantages in expertise, but might look narrow or irrelevant at first glance. That is why your research proposal is so important: it is where you demonstrate which fields you want to focus on and your research criterion and insticts. It is important to keep in mind that a very focused proposal may hit an obvious wall, no matter how well-written it is or how good the candidate: the supervisor may simply not be interested (regardless of area of expertise!) or think that this is the only thing you are interested in and will not consider his suggestions or a possible change of course. Flexibility is quite important, and often disregarded.

It is like fishing in murky waters and so situational that common patterns are difficult to spot. Based on what you write, I would advise you to examine critically if your past applications showed that you were a good match for the project. Did you show relevant skills? Did you identify the important points correctly (e.g. was there a hint of multidisciplinary work that you missed, or did your proposal come across as too narrow)? Did your cover letter and proposal highlighted your strengths with regards to the project? Did you judge the relevance of the supervisors' past research correctly? If you have contact with an academic that can advise you (personal tutor, friendly professor etc) it would be a good idea to discuss the research proposal and the current situation in the field (e.g. you might be "out of fashion", if everyone is currently running behind a certain bandwagon).