Praise be to Allaah.  

The traveller may offer whatever naafil prayers he wants. It is not Sunnah for him to omit any of the naafil prayers apart from the Sunnah prayers before and after Zuhr and the regular Sunnah prayers of Maghrib and ‘Isha’. 

But apart from that the traveller may offer any of the Sunnah prayers. Muslim (680) narrated from Abu Qataadah that the Prophet (peace and blessings of Allaah be upon him) prayed the regular Sunnah prayer of Fajr when travelling. 

And it was also proven that the Prophet (peace and blessings of Allaah be upon him) prayed Duha with eight rak’ahs in Makkah on the day when Allaah enabled him to conquer Makkah. Narrated by al-Bukhaari (357) and Muslim (336). 

And the Prophet (peace and blessings of Allaah be upon him) prayed Witr when travelling. Narrated by al-Bukhaari, 1000. 

Shaykh Ibn ‘Uthaymeen said:  

With regard to the regular Sunnah prayers, I have studied what was narrated in the Sunnah concerning the naafil prayers, and it seems to me that the regular Sunnah prayers of Zuhr, Maghrib and ‘Isha’ need not be offered (by the traveller), but he may offer other naafil prayers apart from that, such as the Sunnah of Fajr, the Sunnah of Witr, qiyaam al-layl, Duha and tahiyyat al-masjid (two rak’ahs to “greet the mosque), and naafil prayers in general. 

Majmoo’ Fataawa Ibn ‘Uthaymeen, 15/258. 

Ibn ‘Umar (may Allaah be pleased with him) criticized those who prayed regular Sunnah prayers when travelling. Al-Bukhaari (1102) and Muslim (689) narrated that Hafs ibn ‘Aasim ibn ‘Umar ibn al-Khattaab said: Ibn ‘Umar set out in the morning on the way to Makkah, and led us in praying Zuhr, two rak’ahs. Then he moved away and we moved away with him until he came to his mount. He sat down and we sat with him. Then he turned back and saw some people standing and said: “What are these people doing?” I said: “They are praying naafil.” He said: “If I had wanted to pray naafil, I would have offered the prayer in full. O son of my brother, I accompanied the Messenger of Allaah (peace and blessings of Allaah be upon him) on a journey and he did not do more than two rak’ahs (when travelling) until Allaah took him (in death). And I accompanied Abu Bakr and he did not do more than two rak’ahs (when travelling) until Allaah took him (in death). And I accompanied ‘Umar and he did not do more than two rak’ahs (when travelling) until Allaah took him (in death). Then I accompanied ‘Uthmaan and he did not do more than two rak’ahs (when travelling) until Allaah took him (in death). And Allaah says (interpretation of the meaning):

 ‘Indeed in the Messenger of Allaah (Muhammad) you have a good example to follow’

[al-Ahzaab 33:21].”.