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We consider an area-minimizing integral current of dimension $m$ andcodimension at least $2$ and fix an arbitrary interior singular point $q$ whereat least one tangent cone is flat. For any vanishing sequence of scales around$q$ along which the rescaled currents converge to a flat cone, we define asuitable singularity degree of the rescalings, which is a real number biggerthan or equal to $1$. We show that this number is independent of the chosensequence and we prove several interesting properties linked to its value. Ourstudy prepares the ground for two companion works, where we show that thesingular set is $(m-2)$-rectifiable and the tangent cone is unique at$\mathcal{H}^{m-2}$-a.e. point.

The fine structure of the singular set of area-minimizing integral currents I: the singularity degree of flat singular points