What are all the nonsplit Lie (and topological) group extensions $0 \to \mathbb{R} \to G \to \mathbb{R}^2 \to 0$? Here, $\mathbb{R}$ and $\mathbb{R}^2$ are regarded as Lie (and topological) groups with respect to the usual addition. One example of a nonsplit extension is the Heisenberg group $H_3(\mathbb{R})$ (Please see a post by Alain Valette at http://mathoverflow.net/questions/63630). Since, every abelian topological extension of $\mathbb{R}^n$ by a locally compact abelian group is trivial, we have that every abelian topological extension of $\mathbb{R}^2$ by $\mathbb{R}$ is trivial. Hence, we need to see only nonabelian extensions.

Central extensions $$ 0 \to \mathbb{R} \to G \to \mathbb{R}^2 \to 0 $$ in which $G$ is a principal $\mathbb{R}$bundle over $\mathbb{R}^2$ (I suppose you mean that by "topological") are classified by continuous maps $$ f: \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R} $$ satisfying $$ f(x,y)f(y,z) = f(x,z). $$ The abelian ones are those corresponding to maps with $f(x,y) = f(y,x)$. This follows from a general theory for topological central extensions described in J.L. Brylinksi's "Differentiable cohomology of gauge groups" (for the smooth case, but that is not relevant) combined with the fact that every principal bundle over $\mathbb{R}^2$ is trivializable. EDIT: From a map $f$, you get the extension $G$ explicitly as the topological space $G = \mathbb{R} \times \mathbb{R}^2$ with the multiplication given by $$ (a_1,x_1)(a_2,x_2) = (a_1 + a_2 + f(x_1,x_1^{1}),x_1 + x_2). $$ 


This is just an answer to the request for the Bianchi classification, not to the original question. I'm putting it as an answer because it's too long for a comment. A 3dimensional Lie algebra $L$ is either semisimple, in which case it is isomorphic to either ${\frak{so}}(3)$ or ${\frak{sl}}(2,\mathbb{R})$, or else it has a basis $x_1,x_2,x_3$ such that $$ [x_1,x_2]=0\qquad [x_2,x_3] = b_{11} x_1 + b_{12}x_2\qquad [x_3,x_1] = b_{21} x_1 + b_{22}x_2 $$ where the $2$by$2$ matrix $B = (b_{ij})$ is equal to one of the following $$ \begin{pmatrix}0&0\cr 0&0\end{pmatrix},\ \begin{pmatrix}1&0\cr 0&0\end{pmatrix},\ \begin{pmatrix}1&0\cr 0&1\end{pmatrix},\ \begin{pmatrix}1&0\cr 0&1\end{pmatrix} $$ or $$ \begin{pmatrix}0&1\cr 1&0\end{pmatrix},\ \begin{pmatrix}1&1\cr 1&0\end{pmatrix},\ \begin{pmatrix}\sigma&1\cr 1&\sigma\end{pmatrix},\ \begin{pmatrix}\sigma&1\cr1&\sigma\end{pmatrix} $$ where $\sigma>0$ is a real number. These are all pairwise nonisomorphic. The proof is fairly straightforward and can be found in many places. 

