<p>The paper concerns front propagation for the following mono-stable reaction-diffusion-advection equation <Equation ID="Equ45"> <EquationSource Format="TEX">\(f(u)u_x + g(u)u_t = [d(u)|u_x|^{p-2} u_x]_x+ \rho (u), \quad (x,t)\in \mathbb {R}\times [0,+\infty ).\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>+</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">[</mo> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>u</mi> <mi>x</mi> </msub> <msub> <mrow> <msup> <mo stretchy="false">|</mo> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo stretchy="false">]</mo> </mrow> <mi>x</mi> </msub> <mo>+</mo> <mi>ρ</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>×</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation>Besides existence and non-existence results for wavefronts solutions, the main focus is their classification: we provide criteria to establish if they attain one or both the equilibria at a finite point and in this case, if they are continuable as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-solutions or if they are sharp solutions.</p>