Here's the first of a couple...
If the FA Cup Winners qualified...
The 2nd/3rd/4th CL spot goes to the FA Cup winners. If the FA Cup winners are already top 1/2/3, the last CL spot instead goes to the highest-placed side not to qualify by league position.
CL qualifiers by year:
1996/97: Man Utd (C), Chelsea (FA)
1997/98: Arsenal (C+FA), Man Utd (2nd)
1998/99: Man Utd (C+FA), Arsenal (L), Chelsea (3rd)
1999/00: Man Utd (C), Arsenal (L), Chelsea (FA)
2000/01: Man Utd (C), Arsenal (L), Liverpool (FA)
2001/02: Arsenal (C+FA), Liverpool (L), Man Utd (L), Newcastle (4th)
2002/03: Man Utd (C), Arsenal (L+FA), Newcastle (L), Chelsea (4th)
2003/04: Arsenal (C), Chelsea (L), Man Utd (L+FA), Liverpool (4th)
2004/05: Chelsea (C), Arsenal (L+FA), Man Utd (L), Everton (4th), Liverpool (CL)
2005/06: Chelsea (C), Man Utd (L), Liverpool (L+FA), Arsenal (4th)
2006/07: Man Utd (C), Chelsea (L+FA), Liverpool (L), Arsenal (4th)
2007/08: Man Utd (C), Chelsea (L), Arsenal (L), Portsmouth (FA)
2008/09: Man Utd (C), Liverpool (L), Chelsea (L+FA), Arsenal (4th)
2009/10: Chelsea (C+FA), Man Utd (L), Arsenal (L), Tottenham (4th)
2010/11: Man Utd (C), Chelsea (L), Man City (L+FA), Arsenal (4th)
2011/12: Man City (C), Man Utd (L), Arsenal (L), Chelsea (FA)
2012/13: Man Utd (C), Man City (L), Chelsea (L), Wigan (FA)
2013/14: Man City (C), Liverpool (L), Chelsea (L), Arsenal (FA)
C: Qualified as Champions
L: Qualified via league position
FA: Qualified as FA Cup Winners
2nd/3rd/4th: Qualified by virtue of FA Cup winners having already qualified.
CL: Qualified as European Champions