I think that the United States stock market will continue to run.
The collapse of the foreign sovereign debt market has made U.S. treasuries relatively more attractive. In addition, bond investors will be looking to U.S. companies with high credit ratings and low debt/equity ratios instead of sovereign debt in general.
This could have long lasting effects, and set off a reflexive chain of events. As investors purchase the bonds of U.S. companies, the yields on these bonds will decrease, making it much easier to sustain larger levels of debt than it has been in the past.
Further, the larger amounts of credit available to companies will allow them to grow at a faster rate. Thus, the companies can shore up their balance sheets and issue new bonds at the same credit rating to continue to grow.
The relationship with stocks will also be reflexive. As the companies grow, their stock prices will also increase. This will decrease the debt/equity ratios of the companies, and make the companies bonds look even less risky.
This situation depends on the companies in question being able to grow their earnings using their credit. Thus, we should look for companies that have growth opportunities, but have been limited thus far because they have not been willing to issue new debt.
Some companies that come to mind are TMO, Thermo-Fisher Scientific, which was one of the first companies to issue new debt at low interest rates following the U.S. downgrade, and CAT, Caterpillar, which relies on high debt loads to fund their manufacturing. CAT is currently sitting on a large backlog of orders.
These are not recommendations necessarily, but they are starting points for further research.
This situation could grow untenable if the companies have no further room to grow. I believe that a collapse in many non-Asian emerging markets is imminent, as a result of the contraction of European credit. Ironically, this could spark a short term rise in U.S. stocks in early 2012, as money managers to pull funds out of riskier emerging markets, like Eastern Europe, and put them into more secure growth stories, like China, or value stocks in America.
But longer term, this may hurt American companies that do business overseas, by contracting their growth opportunities. Further, the high U.S. unemployment will eventually limit U.S. consumers ability to spend, unless new solutions are found. Thus, the situation on a longer term than 1-2 years may not continue, but I believe that the stock market can continue to rise for the near future, on a debt-fueled growth spree.