Darts meets Codegolf
MATLAB (299 249 241 chars)

This is my first serious golfing. My first attempt (136 chars) gives the correct result, but not with the correct formatting. It gives all possibilities looking at the number of points for each dart. This means that single 20 and double 10 do have a separate entry, however, they are both displayed as 20. Of course the last dart is always a double. 

function f(x);u=[1:20 25].';y=[u;2*u; 3*u(1:end-1)];v=combvec([combnk(y,2);[y y];[zeros(62,1) y];[0 0]].',y(22:42).').';v(sum(v,2)==x,:)


In the second attempt the formatting is improved, which has of course increased the number of characters:

function f(x);h=.1;u=h+[1:20,25].';y=[u;2*u;3*u(1:20)];v=combvec([combnk(y,2);[y,y];h*ones(62,1),y];[h,h]].',y(22:42).').';t='SDT';r=@fix;strrep(arrayfun(@(x)[t(int8((x-r(x))/h)),num2str(h*r(x)/(x-r(x)))],v(sum(r(v),2)==x,:),'un',0),'S0','')


Improved from 299 to 249 characters, while at the same time even improving the output formatting. For this improved version the output for the example cases is:

f(170):

'T20'    'T20'    'D25'


f(6):

'S1'    'S3'    'D1'
'S1'    'T1'    'D1'
'S2'    'D1'    'D1'
'S2'    'S2'    'D1'
'D1'    'D1'    'D1'
''      'S4'    'D1'
''      'D2'    'D1'
'S1'    'S1'    'D2'
''      'S2'    'D2'
''      'D1'    'D2'
''      ''      'D3'


f(169):

Empty cell array: 0-by-3


Additional:

According to my calculation skills there are a grand total of 42336 possibilities to end the dart game.