Abstract We study global well-posedness and finite time blow-up of solutions for a nonlinear one-dimensional transport equation with nonlocal velocity <?CDATA ${u}_{t}-{\left(\mathcal{H}\left(u\right)u\right)}_{x}=\nu {u}_{xx}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mrow> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi mathvariant="script">H</mml:mi> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> </mml:mfenced> <mml:mi>u</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>ν</mml:mi> <mml:msub> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> </mml:math> , ν &gt; 0, and measure initial data. Such model arises in fluid mechanics in vortex-sheet problems and its nonlocal feature comes from the presence of a singular integral operator (Hilbert transform <?CDATA $\mathcal{H}\left(u\right)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">H</mml:mi> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> </mml:mfenced> </mml:math> ) in the velocity field. In the viscous case ν &gt; 0, we analytically obtain an explicit condition on the size of the initial data for the global well-posedness in the framework of pseudomeasure spaces. In fact, we can give the condition depending on the initial-mass and analyze how the flow evolves from singular measures. Also, we numerically study blow-up of concentration type and global diffusion-smooth behavior of solutions. We obtain numerics that indicate the threshold value 2 π for the initial-data mass that decides between blow-up or global smoothness of solutions. Such value is the same obtained for regular initial-data and by means of entropy methods. Thus, it seems to be intrinsic to the nonlocal PDE and independent of a particular framework, approach and initial-data regularity. The inviscid case ν = 0 is remarkable: simulations for model <?CDATA ${u}_{t}-{\left(\mathcal{H}\left(u\right)u\right)}_{x}=0$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mrow> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi mathvariant="script">H</mml:mi> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> </mml:mfenced> <mml:mi>u</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> , evidence that the solution presents blow-up of concentration type with mass-preserving, while an attenuation effect is observed for the model with opposite sign in the nonlinearity <?CDATA ${u}_{t}+{\left(\mathcal{H}\left(u\right)u\right)}_{x}=0$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mrow> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi mathvariant="script">H</mml:mi> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> </mml:mfenced> <mml:mi>u</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> , for any nontrivial (positive) measure as initial data.