If I wake up after sunrise, shall I only do the obligatory morning prayer or start with the Sunnah (supererogatory) prayer and then the obligatory?

           In the name of Allah. May Allah's Blessing and Peace be upon His Messenger Muhammad.

A Muslim should be careful to perform the prayer in its prescribed time. Allah says, "Prayers are obligatory for the believers at prescribed times". (Qur'an, 4: 103) A Muslim must take all the means that would make him perform his/her prayers in their due time. If s/he wants to pray Fajr (Dawn) in its time, then he must sleep early with the intention of praying asking someone to wake him up, or use an alarm clock for the due time.

If he hears the call to Fajr, he must rise from his bed and rush to his Lord, so that he may start his day by obedience and diligence, as the Prophet (Peace be upon him) said, "Satan ties three knots at the back of your head when you sleep, and he seals the place of each knot with saying, 'You have a long night ahead, so sleep'. If you wake up and mention Allah's name, a knot is untied. If you do wudūˀ (ablution), another knot is untied. If you pray, the last knot is untied, and your morning will let you lively and in good spirits. If not, your morning shall leave you in bad spirits and laziness". (Hadith reported by both Bukharῑ and Muslim)

If he does not hear the call, after taking all above-mentioned necessary means, and wakes up after the sun has already risen, s/he may then make up for it, and there would be no sin on him in this case. The Prophet (pbuh) said, "Whomsoever forgets [to perform] a prayer or missed it due to sleep, its expiation is that he should perform it whenever he remembers". (Hadith reported by both Bukharῑ and Muslim) Allah says, "And keep up with the prayers so that you may remember Me". (Qur'an, 20:14)

Now we come to the question of how to pray Fajr after its prescribed time has elapsed and after the sun has already risen. According to Prophet’s act, a person should start with the two Sunnah rakˁahs, then he can offer the (obligatory) prayer, as making up for a missed prayer should be in the same as the original one. This is how the Messenger of Allah (pbuh) did when he missed Fajr prayer during one of his travels, and neither he nor any of his companions woke up till after sunrise.

Abu Qatādah reported that "The Prophet (pbuh) was on a journey. He said to his companions, 'Watch the time for our prayer, i.e. the Fajr prayer'. But sleep dominated them all, and they were awakened only with the heat of the sun. So, they did their ablution. Bilal then called for the prayer and they offered two Sunnah rakˁahs of Fajr and then offered the Fajr prayer and mounted their animals. Some of them said to others, 'We showed negligence in our prayers'. The Prophet said, 'There is no negligence in sleep. The negligence can only be when you are awake. If any of you misses a prayer, he should offer it when he remembers it'". (Reported by Abu Dawoud)

In this Hadith, there is an indication that the Sunnah of Fajr could be offered before the obligatory, because the Prophet (pbuh) commanded Bilal to call for prayer and then he (pbuh) offered two Sunnah rakˁahs of Fajr. Then he ordered for the main obligatory prayer to be held after that, and prayed it.

Allah knows the best.