According to an online dictionary (spanishdict.com), one can say "I'm sorry" in three ways:
If I want to say "I'm sorry, but I can't help you", which verb would be best to use?
"Disculpe" is used more when you want to ask something. If you want to be polite, talking to someone you don't know you can say:
-Disculpe, me podría decir la hora, por favor.
-Sorry, could you tell me, please, what time is it?
You can say it too with "Perdón" or "Perdone" instead of "Disculpe"
"Lo siento" is allways to apologize.
Adding more information, you can say "disculpa" (excuse me) or "perdón" (pardon) when interrupting a conversation, but not "lo siento" (because you don't feel sorry for interrupting that conversation).
On the contrary, you can say "lo siento" at a funeral, but not "disculpa" or "perdón" (because it's not your fault the death of that person).
"Lo siento" means "I feel it". Therefore can be used when you apologize for a mistake that you've made that make someone else feel bad. But it can also be used to express sympathy for something that is not your fault (a friend got sick for example).
"Disculpe" would be used as "excuse-me" when trying to pass someone in a crowd for example or to interrupt someone for a question.
"Perdón" is asking for personal forgiveness. It could be for little issues like if you accidentally bumped someone or if you want someone to repeat something you did not hear correctly (or were not paying attention to). Or it could be used to ask forgiveness to more serious offenses.
Perdón is mostly used when you made a huge mistake and you regret it from the bottom of your heart. But if you want to say some like you say, it's enough with lo siento or disculpa.
The difference in meaning is not that big, except the last one:
Lo siento, pero no puedo ayudarte.
Disculpa, pero no puedo ayudarte.
Perdón, pero no puedo ayudarte.
The last one normally requires a little bit of emphasis:
Perdón, pero realmente no puedo ayudarte.
Now disculpa has a different use when you offer it:
Te debo una disculpa.
However, we don't use lo siento as follows:
Te debo un lo siento.
I fear I may sound totally subjective here, but to me:
"Lo siento" is the literal translation of "I'm sorry"
"Disculpa" means you acknowledge there has been a fault, but it wasn't your fault", like when you bump into someone in the train.
"Perdón" means you know it's your fault and you ask for forgiveness.
