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 a connected graph $G$ has $n$ vertices and maximum degree~$d$ and admits an identifying code, 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 constants, 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, $\M(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n$. Using the former result, we give sharp estimates for the size of the minimum identifying code of random $d$-regular graphs, which is about $\tfrac{\log d}{d}n$.