This sounds like an ordinal regression problem. Logistic regression is model for binary data, with more then two categories, you can use multinomial regression, or (if this makes any sense) treat them as continuous and use linear regression. The problem with multinomial regression for such data would be that it doesn't care about ordering of the categories, and models all the categories as separate entities. This may be a problem, for example, in cases like modelling Likert-style data in form of (1) I strongly disagree - (2) I disagree - (3) I have no opinion - (4) I agree - (5) I strongly agree, where you implicitly assume that there are are some incremental relationships between the answers (if someone strongly agrees, then she agrees as well and doesn't disagree etc.). What follows, in such cases people sometimes treat such variables as continuous and use linear regression, but the problem wit linear regression is that it assumes that your categories have numerical meaning, and arithmetic operations on them are also meaningful (agree + 1 = agree * 1.25 = strongly agree).
Because the standard regression models do not really work for such "incremental" data, there is a special model for it, the ordinal regression model. It assumes that your outcome is on ordinal scale, so the order of the categories matters, but they have no numerical meaning. This seems to be your case, as there seems to be obvious order: no event < non-severe event < severe event. It models cumulative probabilities
\Pr(Y \le i|X) = \sigma(\theta_i - X\beta)
where $\sigma$ is the links function (logit, or probit etc.). For more details, you can check also other questions tagged as ordered-logit.