The authors study the action of a \(p\)-group \(G\) on a polynomial ring over a field of characteristic~\(p\) by linear transformations of the variables. The main goal is to find methods for the symbolic computation of generators for the ring of invariants under such an action. As a kind of appetizer, the authors start by showing that with a suitable (and natural) choice of variables the ring of invariants always has a finite SAGBI basis. The authors continue by presenting a minimal system of generators for the vector invariants of the cyclic group of order~\(p\) acting non-trivially in dimension two, by extracting this from a generating system given by \textit{H. E. A. Campbell} and \textit{I. P. Hughes} [Adv. Math. 126, No. 1, 1--20 (1997; Zbl 0877.13004)]. The main part of the paper contains the development of two new methods for computing invariants. One of them uses a test whether a ``candidate'' ring coincides with the ring of invariants by comparing suitable localizations. The other one (called the ladder algorithm) is an iterative approach using a composition series of \(G\). Interestingly, the ladder method requires extensive computations in some cohomology module. The paper contains an interesting application: The authors consider the group \(U_3(p)\) of \(3 \times 3\) upper unipotent matrices over \(\mathbb{F}_p\) acting on two copies of the natural module. Using the ladder method, the authors manage to compute a full system of generating invariants for \(p = 3\). For \(p = 2\) the computation is much easier. For other values of~\(p\) the authors carry the computations far enough to be able to show that the ring of invariants is not Cohen-Macaulay.