No particular individual dataset of observations definitively 'proves' the correlation of human caused climate change. But the cross correlations between an incredibly diverse set of observations does provide a basis for a pretty well based assumption. These include ice cap bores from Greenland, species in seabed sediments, coral growth, tree rings (even petrified trees
What hobbles this in every case is the sample space, what is needed a fine resolution chronographic continuous globally distributed climate record. I.e. the ice caps are only located in certain areas and so it has only been in the last few years we have had satellite platforms to global measure sea temperatures. Global weather monitoring on a regular basis only started during World War Two, mostly driven by military aviation.
If Climatology is a tough nut, Paleoclimatology is even tougher. Ironically, the world wide exploration for oil combined with temperature as an indicator for petroleum formation has provided one possible set of observations. See " Optimal Surface Temperature Reconstructions Using Terrestrial Borehole Data" (and others) at http://www.ncdc.noaa.gov/paleo/pubs/mann2003/mann2003.html This area of current relies on ancillary data, but could be extended to deeper wells and better distributed locations to increase the time horizons.
The diverse and broad studies around climate change complement and supplement one another to reach the conclusion and correlation. Sunspots and lemming migrations can be argued endlessly, but it is the meta analysis of all these efforts that matters. If the media has a hard time with translating and portraying the problem and controversy inside a particular specialized scientific study, it is absolutely incapable of informing the public about the meta analysis. So drowning cute polar bears isn't scientifically precise, if the imagery causes behavior change, all the better.
The other aspect of the debate is the time dynamics and values of the risk situation: What is the cost of doing business as usual in the event the warming hypothesis is wrong? If we mitigate the carbon impacts and it's wrong, so what? We have a vastly more efficient and clean economy. If it's right, the downside is potentially death and disruption for billions. Also, how long do we have to figure it out?