The linear support vector machine has a parametrised decision boundary. The paper considers inference for the corresponding parameters, which indicate the effects of individual variables on the decision boundary. The proposed inference is via a convolution-smoothed version of the SVM loss function, this having several inferential advantages over the original SVM, whose associated loss function is not everywhere differentiable. Notably, convolution-smoothing comes with non-asymptotic theoretical guarantees, including a distributional approximation to the parameter estimator that scales more favourably with the dimension of the feature vector. The differentiability of the loss function produces other advantages in some settings; for instance, by facilitating the inclusion of penalties or the synthesis of information from a large number of small samples. The paper closes by relating the linear SVM parameters to those of some probability models for binary outcomes.