Signatures have started out as an honor system: before widespread literacy, there was no realistic means of telling one X from another. They haven't come far since.
Currently, the only security measure is that ink signatures on paper can be analyzed by forensic document examiners to determine if a signature matches others produced by the person it's meant to identify. This requires physical access to the paper to examine the ink very closely.
Translating this to modern security concepts, there is no secret in the image of the signature. Rather, the image is public and simultaneously serves as a signature and a sample (certificate) for verifying other signatures by the same person. The secret or the private key is the exact sequence of strokes with the angle, velocity and pressure for each, required to produce the same imprint on paper.
These parameters and their distribution over each stroke are difficult to match manually and will usually differ enough for an expert to distinguish. Non-manual reproduction, be it by photocopy or facsimile, is much easier to detect.
Electronic tablets meant to capture handwritten signatures digitally also record and store these parameters, not just the image, in an attempt to provide similar protection. However, the current industry practice for this, at least in the largest vendor's implementation, relies on steganography rather than cryptography, rendering it insecure.