A spec sheet lists min, typical, and max. Which do you design to and why?

You design to the min/max worst case, not the typical, because "typical" is a nominal value at nominal conditions that is explicitly not guaranteed, it's representative of an average part under average conditions, but real parts vary across manufacturing lots, temperature, and supply voltage, and any individual part you ship could be at a corner. So you use whichever bound represents the worst case for your design margin: the minimum guaranteed output current when sizing a load (don't assume the typical), the maximum oscillator drift when computing whether a UART baud rate stays within tolerance, the maximum startup or settling time when deciding how long to wait, the minimum guaranteed logic-high threshold when checking signal compatibility. Designing to typical means a corner part, perfectly within spec, violates your assumptions and the product fails in the field intermittently or at temperature extremes, which is among the hardest classes of bug because it's lot- and environment-dependent. The min/max values are what the manufacturer guarantees; the typical is informational. The corollary is that you must also read the conditions attached to each spec, since a min/max is only valid under its stated temperature, voltage, and load, a number quoted out of its conditions is meaningless. So: design to guaranteed min/max worst case, with attention to the conditions footnotes, and use typical only for rough estimation.