The general goal of the paper is to prove versions of known approximation results on foliations of codimension one on \(3\)-manifolds with lower order of differentiability. Let \(\mathcal F\) be a foliation of a smooth \(3\)-manifold \(M\) (recall that any topological \(3\)-manifold has a smooth structure, which is unique up to diffeomorphisms). \(M\) may have boundary or corners; in this case, it is assumed to be sutured in the sense of Gabai. Roughly speaking, for \(k\ge l\) in \(\mathbb N\cup\{\infty\}\), the notation \(C^{k,l}\) is used to indicate that \(\mathcal F\) is tangentially \(C^k\) and transversely \(C^l\). All foliations of the paper are assumed to be at least \(C^{1,0}\); in particular, this applies to \(\mathcal F\) in the statement of the main theorems. Then \(C^0\) close foliations means that the fields of tangent planes to their leaves are \(C^0\) close. The first main theorem of the paper states that there is a small \(C^0\) isotopy of \(M\) taking \(\mathcal F\) to a \(C^{\infty,0}\) foliation that is \(C^0\) close to \(\mathcal F\). The second main theorem states that, if \(\mathcal F\) is also transversely oriented and measured (endowed with a continuous transverse measure), then there is an isotopy of \(M\) taking \(\mathcal F\) to a \(C^\infty\) measured foliation that is \(C^0\) close to \(\mathcal F\). As a corollary it follows that \(\mathcal F\) is \(C^0\) close to a smooth fibration over \(S^1\) (a version of the Tischler’s theorem). The last main theorem states that \(\mathcal F\) can be arbitrarily \(C^0\) approximated by a \(C^{\infty,0}\) Denjoy blowup along any prescribed countable collection of leaves. The main tool used in the proofs is a certain kind of flow box decomposition introduced by the authors.