For plane curves, general sufficient conditions have been given by Shustin (Trans. AMS 356, 2004, 953–985) although for particular singularity types (such as A-singularities) sharper results are known (see J. Alg. 302, 2006, 37-54). For one single $A_m$ singularity, I think the best sufficient condition is due to Lossen, via explicit equations (I can't find the reference right now). In general it is not enough that the linear system of plane curves of degree $d$ has dimension at least equal to the codimension of the singularity type (except for the case of $m$ nodes, when this is necessary and sufficient).

In higher dimension, less is known, but again I'd suggest to look at Shustin-Westenberger, J. London Math. Soc. 70, 609–624.

For plane curves, general sufficient conditions have been given by Shustin (Trans. AMS 356, 2004, 953–985) although for particular singularity types (such as A-singularities) sharper results are known (see J. Alg. 302, 2006, 37-54). For one single $A_m$ singularity, I think the best sufficient condition is due to Lossen, via explicit equations (EDIT: Comm. Algebra 27, 1999, 3263–3282). In general it is not enough that the linear system of plane curves of degree $d$ has dimension at least equal to the codimension of the singularity type (except for the case of $m$ nodes, when this is necessary and sufficient).

In higher dimension, less is known, but again I'd suggest to look at Shustin-Westenberger, J. London Math. Soc. 70, 609–624.