Let X= (Xt)t more than or equal to 0 to be a real-valued stochastic process which is self-similar with exponent H>0 and has stationary increments. Several results about the marginal distribution of X1 are given. For H inequal to 1, there is a universal bound, depending only on H, on the concentration function of logXsuper+sub1. For all H>0, X1 cannot have any atoms except in certain trivial cases. Some lower bounds are given for the tails of the distribution of X1 in case H>1. Finally, some results are given concerning the connectedness of the support of X1.