Since the case for AGW depends on many different threads of evidence, the cases against that evidence are numerous. For example:...
Proponents of the AGW theory claim that recently observed warming is unprecedented. Against this, critics offer the Roman warm period and the Medieval warm period. Proponents assert that the Medieval warm period either did not happen or was localized to Northwest Eurasia (Europe). Against this claim are studies of lake sediments in China, South America, and (iirc) the American Southwest that find cooling during the European Medieval warm period.
Some proponents of AGW (Mann, et. al.) used tree-ring data as proxy thermometers in constructing the hockey-stick graph. Steve McIntyre demonstrated that one of these reconstructions depended on very few samples and ignored an available, wider sample from the same area that moved the trend in the opposite direction (the "Yamal chronology", apparently the people who used the data cherry-picked it to get the trend they wanted to see). Against the AGW claim that tree rings (of some species) make reliable thermometer proxies, skeptics make several objections: "climate", as assessed by a few ancient tree trunks buried in a sandbank, may be quite local, and for the time that tree lived, locally unusally warm or cold, or wet or dry. A herd of reindeer might have fertilized them, or stunted them with salt. Etc. Sample size matters, here, as do assumptions about the relation between ring width and other variables (soil chemistry, water, etc.) for various species.
Trakar observed that many disciplines contribute to this discussion (at a professional level). No one has all the relevant expertise. Experts in one discipline have to trust experts in other disciplines for their analyses. Experts often have to trust other experts in their own discipline when it comes to collection of raw data, since field work is time-consuming and no one can collect buried trees and measure tree rings from every continent from beds dating back to the end of the last Ice Age, and no one can personaly monitor every thermometer and remote sensor that supplies current temperature data. People who use the raw data have to trust the field workers who supply that data. With their behavior as revealed in the leaked UEA emails and elsewhere, Keith Briffa,
Phil Jones, Michael Mann, Steven Schneider, and Kevin Trenberth have forefeited that trust.