<p>The <Emphasis FontCategory="SansSerif">SHA-2</Emphasis> family is a U.S. federal standard published by NIST. Due to its complex design compared with <Emphasis FontCategory="SansSerif">SHA-1</Emphasis>, there is almost no progress in collision attacks on <Emphasis FontCategory="SansSerif">SHA-2</Emphasis> after ASIACRYPT 2015. In this work, we retake this challenge and aim to significantly improve collision attacks on the <Emphasis FontCategory="SansSerif">SHA-2</Emphasis> family. First, we observe from many existing attacks on <Emphasis FontCategory="SansSerif">SHA-2</Emphasis> that the current advanced tool to search for <Emphasis FontCategory="SansSerif">SHA-2</Emphasis> characteristics has reached its limits. Specifically, longer differential characteristics could not be found and collision attacks on more steps could not be reached. To overcome this obstacle, we adopt Liu et al.’s MILP-based method for signed differential characteristics published at EUROCRYPT 2023, and implement it with SAT/SMT for <Emphasis FontCategory="SansSerif">SHA-2</Emphasis>, where we also add more techniques to detect contradictions in <Emphasis FontCategory="SansSerif">SHA-2</Emphasis> characteristics. This answers an open problem left in Liu et al.’s paper to apply the technique to <Emphasis FontCategory="SansSerif">SHA-2</Emphasis>. As a result, we successfully mount the first practical collision attack on 31-step <Emphasis FontCategory="SansSerif">SHA-256</Emphasis> and semi-free-start (SFS) collision attack on 39-step <Emphasis FontCategory="SansSerif">SHA-256</Emphasis>, respectively. In addition, we also report the first practical free-start (FS) collision attack on 40-step <Emphasis FontCategory="SansSerif">SHA-224</Emphasis>, improving the best theoretic 40-step attack of time complexity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2^{110}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mn>110</mn> </msup> </math></EquationSource> </InlineEquation>. Moreover, we can also mount practical and theoretic collision attacks on 28-step and 31-step <Emphasis FontCategory="SansSerif">SHA-512</Emphasis>, respectively, which improve the best collision attack only reaching 27 steps of <Emphasis FontCategory="SansSerif">SHA-512</Emphasis> at ASIACRYPT 2015. Since we can search for longer differential characteristics, the collision attacks on <Emphasis FontCategory="SansSerif">SHA-2</Emphasis> in the quantum setting are also improved. In summary, this paper achieves some notable progress in the analysis of <Emphasis FontCategory="SansSerif">SHA-2</Emphasis> after the major achievements made at EUROCRYPT 2013 and ASIACRYPT 2015.</p>