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We prove that the Beltrami operator on closed oriented 3-manifolds, i.e. thesquare root of the Hodge Laplacian on its coclosed spectrum, generically has1-dimensional eigenspaces, even along !-parameter families of $\mathcal{C}^k$Riemannian metrics, where $k\geq 2$. We show further that the Hodge Laplacianin dimension 3 has two possible sources for nonsimple eigenspaces along generic1-parameter families of Riemannian metrics: either eigenvalues coming frompositive and from negative eigenvalues of the Beltrami operator cross, or aclosed and a coclosed eigenvalues cross. We provide examples for both of thesephenomena. In order to prove our results we generalize a method of Teytel,allowing us to compute the meagre codimension of the set of Riemannian metricsfor which the Beltrami operator and the Hodge Laplacian have certain eigenvaluemultiplicities. A consequence of our results is that while the simplicity ofthe spectrum of the Hodge Laplacian in dimension $3$ is a meagre codimension 1property with respect to the $\mathcal{C}^k$ topology (as proven by Enciso andPeralta-Salas), it is not a meagre codimension 2 property.

In Dimension 3 the Beltrami Operator has simple Spectrum along generic 1-Parameter Families of Riemannian metrics, but the Hodge Laplacian does not