I've not designed a tall building myself -- so I'm curious to hear others contribute as well. But when you say global stability, my mind goes three places:
1. Sliding and overturning checks. Thankfully, buildings (essentially) never slide, and rarely have overturning issues. (I'd only ever give it significant thought for very high lateral loads combined with unique foundation systems).
2. Checking lateral deflections, which could affect at least the following:
Absolute deflection limits (for buildings near property lines)
Story drift limits for serviceability and architectural elements
P-Delta effects on your structural elements
P-Delta stability during construction (depending on construction sequence)
Also, I'd argue that long-term performance of the foundation system should be considered in part 2, so you don't end up with a Millenium Tower type situation (
3. For extremely tall buildings, I might consider aero-elasticity and dynamic wind/vibration responses to fall under the category of "global stability".
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The name is a long story -- just call me Lo.