Scenario: A user is sending SMS to leads( one-to-one or multiple ) and it is getting delivered as well. But SMS sent activity is not getting posted against the lead.

Reason: A setting had been done by the user which was causing the issue. If we do this: Settings--> Customisation--> Custom activities and scores--> SMS sent--> Update custom activity type-->In step 2, make Owner field mandatory, as there is no option to select owner while sending SMS, the activity will got posted.

Solution: It is recommended not to make  owner field mandatory.