For the query:
SELECT column1 FROM Table1 ORDER BY 'x';

the optimizer will throw out the ORDER BY clause. This is another example of dead code elimination.

For the query:
SELECT column1 FROM Table1 ORDER BY column1;

the optimizer will use an index on column1, if it exists.

For the query:
SELECT column1 FROM Table1 ORDER BY column1+1;

the optimizer will use an index on column1, if it exists. But don't let that fool you! The index is only
for finding the values. (It's cheaper to do a sequential scan of the index than a sequential scan of the table,
that's why index is a better join type than ALL — see Section 3.2.2.4, “The index Join Type”.)
There will still be a full sort of the results.

For the query:
SELECT * FROM Table1
WHERE column1 > 'x' AND column2 > 'x'
ORDER BY column2;
i

f both column1 and column2 are indexed, the optimizer will choose an index on ... column1. The
fact that ordering takes place by column2 values does not affect the choice of driver in this case.
See: /sql/sql_select.cc, test_if_order_by_key(), and /sql/sql_select.cc,
test_if_skip_sort_order().

ORDER BY Optimization [http://dev.mysql.com/doc/refman/5.1/en/order-by-optimization.html],
provides a description of the internal sort procedure which we will not repeat here, but urge you to read,
because it describes how the buffering and the quicksort mechanisms operate.
See: /sql/sql_select.cc, create_sort_index().