Abstract: A fundamental concept in geometric measure theory is the division of sets and measures into rectifiable ("regular") and purely unrectifiable ("irregular") pieces. The qualitative theory of rectifiable sets and absolutely continuous rectifiable measures in Euclidean space developed across the last century, beginning with the seminal work of Besicovitch (1928, 1938) and later generalized and improved upon in a series of papers by Morse and Randolph (1944), Moore (1950), Marstrand (1964), Mattila (1975) and Preiss (1987). In particular, in the presence of absolute continuity, these investigations revealed a deep connection between the rectifiability of a measure and the asymptotic behavior of the measure on small balls. A quantitative counterpart to this theory emerged in the 1990s, with major contributions including the work of Jones (1990), David and Semmes (1991, 1993), Okikiolu (1992) and Pajot (1996, 1997). In this talk, I will discuss recent joint work with Raanan Schul. We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in Euclidean space. In particular, we give new necessary conditions for a measure to give full mass to a countable family of finite length curves. A novelty of our main result is that no assumption is made on the upper Hausdorff density of the measure. Thus we are able to analyze general 1-rectifiable measures, including measures which are singular with respect to Hausdorff measure.