There are many free parameters in the commonly used reproducing kernel (“RK”) and moving least squares (“MLS”) meshfree approximations. Among them are the order of polynomial completeness, continuity of the kernel, specific type (e.g., spline or Gaussian) and shape (brick, sphere, ellipsoid, etc.) of kernel, and its measure. For instance, a popular choice is to use a brick-shaped support with normalized measure of 1.5, and cubic B-spline kernel. Yet this choice does not seem well-motivated in the absence of a priori knowledge of the solution in the PDE: generally speaking, for smooth solutions, we would like the kernel to be very smooth, and for rough solutions (common in hard applications), we would like the kernel to be rough. Nevertheless, we often do not know beforehand how smooth or rough, and where in space-time. This also leaves aside the question of what basis to choose, and what specific measure, type, and shape of kernel is best. These are practical questions for analysts who would like to use these very capable approximations, but there exists virtually no guidance other than the very limited studies in the literature. It should be noted there is sometimes a non-trivial cost-error trade-off for these choices. Further motivation for looking into these parameters is the knowledge that these parameters determine C in the error estimates [1], thereby offering a chance to lower the error with better parameters. And, it already known that it possible to lower the order of error by at least and order of magnitude without any additional cost in a meshfree formulation [2]. In this talk, a massive parametric study of the free parameters in the RK/MLS approximation is presented. Many interesting counterintuitive results are obtained, and it is shown that the optimal selection often runs contrary to expert knowledge. REFERENCES [1] Harari I., Hughes, T.J. What are C and h?: Inequalities for the analysis and design of finite element methods. Computer Methods in Applied Mechanics and Engineering, Vol. 97(2), pp.157-192, 1992. [2] Chen J. S., Hillman M., Chi S. W., Meshfree methods: progress made after 20 years, Journal of Engineering Mechanics, Vol. 143(4), pp. 04017001, 2017.