This position is the late 19th century early 20th century idea of point particles as singularities in the continuous field, meaning that they are just points where fields come out of, and they have no internal structure. This idea had its heyday in the late 19th century with the model of the electron as a singularity in the EM field. It is in some ways the modern quantum field picture, but the adjective "quantum" is essential, without it, the picture is completely wrong.
The central problem with the pointlike electron is that the mass of the electron is partly due to the energy in the field it carries around with it, and this energy is divergent in the limit of a point electron, so that the electron would have to have less mass than the energy contained in its electric field. This is a paradox, because it requires that you subtract out a negative mass contribution at the point location of the electron which is formally infinite to restore a finite mass to the electron. Because of this, the equations of motion for the electron become unstable.
The hallmark of this instability is the radiation reaction term, the reaction of the electron to the lightwaves it emits, is proportional to the derivative of the acceleration. Dirac considered how to restore sensible behavior to the classical electron, and showed that if you do so, the electron gets nonlocal forces, so that it will accelerate in response to an applied field slightly before the field gets to it. This behavior is unacceptable, and it is essentially due to the negative mass for a pointlike electron, it is begging for the electron to be extended, and to be at least as big as its classical electron radius.
All these classical ideas are pointless and obsolete today, because in quantum mechanics, the particles are completely different objects, defined by quantum motion of fields, not by the location of classical points (at least not in a causal field picture). The notion of a point particle was replaced by the more subtle notion of a quantum point particle, which has a probability amplitude to be at various places. This quantum point particle can reproduce the quantum field if it is allowed to go backward and forward in time.
The quantum mechanical notion of point particle does not require you to think of it as a singularity of the field, because the process of localization of the particle requires a position measurement, which requires higher energy as you go deeper. This postpones the problems of divergence in the self-mass to much higher energy, and the problem resolves itself because the scale of problematic behavior is pushed beyond the scale where gravity kicks in.
At the scale of gravity, there is the notion that any sufficiently dense object is a black hole. The point particles of classical physics are best replaced by the extended black hole objects of General Relavity. A black hole pre-accelerates in response to a field, but this is not paradoxical, because the black hole is extended, and its horizon is a globally defined object. Black holes can be charged, and classically, the charge is less than the mass.
But quantum mechanically, the quantization of black hole horizon motion, which is string theory, gives light particles whose charge is greater than the mass. The electron in string theory is then viewed as a type of black hole, one with extreme charge, and with a certain spatial extent, and winding certain cycles in space, so that it is massless in the first approximation.
But the classical ideas of a particle as a singularity in the field is only loosely related to the modern picture, and there is no reason to consider it anymore.