Debts which may not be discharged (generally) include:
Of course, there is some flexibility to debts which generally cannot be discharged. For instance, it is an Urban Legend that student loans cannot be discharged. In some cases, they may. If the student loan debt can be demonstrated to be a serious burden even with the discharging of all other debt, then there is the possibility of elimination. This is rare, but does happen.
In addition, Federal Bankruptcy Judges have considerable latitude in determining what is to be eliminated and what is not. If you have liquid assets (money) and file Chapter 7, the Court can force Chapter 13 instead. Chapter 13 allows for careful debt restructuring. The judge considers each case individually and generally attempts to resolve the debt problem in whatever way is best for the individual involved.