You can't change people, and trying to do so may hurt your relationship. You can drop hints, and then leave it up to them to decide!

[edit] Steps

Decide if your significant other knows their clothes are ugly. They may not have thought about their clothes, or that other people think about their appearance.
Know if you are willing to be honest with them, possibly offending them. You may not want to invest time in helping someone with their appearance if it is futile. Can you be honest with them? Do you plan on continuing your relationship with this person?
Buy something aesthetically pleasing that you think they will like. You may not like its style, but if the garment is an improvement, it will be worthwhile.
Go shopping with them and discuss clothing that you like.

[edit] Tips

Between the two of you, decide on an area for you to work on (organizational skills, compliments, etc). Working together to solve your trouble areas can ease conflict.
Try repeating steps 3 and 4 every few months. Over time, you may be able to show your significant other how to amend their style slightly. That is about the best you can do.
Be ready to compromise. If they is willing to accept something new just to please you, then be grateful.
Be sensitive. You should be aware that your significant other may have a strong 'self image', and may only feel comfortable with certain styles of clothing. Do not try to force them to wear something that isn't comfortable, or that they don't like. If you can work within these boundaries to find something that you both like, do so.

Don't try to change people or their habits. Ultimately, the only people we can change are ourselves.
Remember that any definition of 'style' is at least partly subjective, and people should not be valued any less because their definition of what is stylish does not conform to a majority opinion.
You might lose your significant other by coming on too strong. If you are unwilling to be patient, you might as well save the ugly clothes discussion for the breakup.