An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If $\M(G)$ denotes the minimum size of an identifying code of a graph $G$, it was conjectured by F.~Foucaud, R.~Klasing, A.~Kosowski and A.~Raspaud that if $G$ is connected, has $n$ vertices and maximum degree~$d$, then $\M(G)\leq n-\tfrac{n}{d}+O(1)$. We use probabilistic tools to show that $\M(G)\leq n-\tfrac{n}{\Theta(d^3)}$ holds. For $d$-regular graphs and further classes of graphs, this bound is improved to $\M(G)\leq n-\tfrac{n}{\Theta(d^{3/2})}$. Moreover, if $G$ is regular and has no vertices sharing the same neighbours, we show that the conjecture holds up to a constant factor (i.e. $\M(G)\leq n-\tfrac{n}{\Theta(d)}$). In the second part, we prove that in any graph $G$ of minimum degree~$\delta$ and girth~5, $\M(G)\leq\tfrac{3\log\delta}{\delta}n$. Finally, using the former result, we also determine that almost all $d$-regular graphs have their minimum identifying code of size $\Theta\left(\tfrac{\log d}{d}\right)\cdot n$.