The Shia will say that ‘Umar (ra) ran away from the battle of Uhud and they will present the following proofs usually (because there aren’t any other proofs):
حدثنا : أبو هشام الرفاعي ، قال : ، ثنا : أبوبكر بن عياش ، قال : ، ثنا : عاصم بن كليب ، عن أبيه ، قال : خطب عمر يوم الجمعة ، فقرأ : آل عمران ، وكان يعجبه إذا خطب أن يقرأها ، فلما إنتهى إلى قوله : إن الذين تولوا منكم يوم التقى الجمعان ، قال : لما كان يوم أحد هزمناهم ، ففررت حتى صعدت الجبل ، فلقد رأيتني أنزو كأنني أروى ، والناس يقولون : قتل محمد ! فقلت : لا أجد أحداًً يقول قتل محمد إلاّ قتلته ، حتى إجتمعنا على الجبل ، فنزلت : إن الذين تولوا منكم يوم التقى الجمعان
From ibn Jarir al-Tabari in Jami’i al-Bayan 4/193:
Abu Hisham al-Rifa’ee said: Abu Bakr ibn ‘Ayyash said: ‘Issam ibn Kulaib from his Father: ‘Umar made a sermon on Friday so he read Surat aal-‘Imran and he liked to recite it when he made his sermons and when he ended on “Indeed, those of you who turned back on the day the two armies met” He said: On the day of Uhud we defeated them so I fled and climbed the mountain … (until the end of the narration).
Hadith grading: Abu Hisham ibn Yazeed al-Rifa’ee al-Kufi is weak by consensus of the scholars of Hadith, so this narration is extremely weak.
The second narration is that of Ibn Hisham al-Hamiree in Seerat al-Nabi 2/281:
قال الراوي : قال إبن هشام : وكان ضرار لحق عمر بن الخطاب يوم أحد ، فجعل يضربه بعرض الرمح ويقول : إنج يابن الخطاب لا أقتلك !! فكان عمر يعرفها له بعد إسلامه.
Ibn Hisham said: and Dirar pursued ‘Umar ibn al-Khattab on the day of Uhud and he began hitting him with the bottom of the spear…(until the end)
Hadith grading: Between Ibn Hisham and the battle of Uhud are fifty years so this narration is Mursal and weak and it doesn’t mention him running away.
These are pretty much the only narrations that mention anything about ‘Umar ibn al-Khattab (ra), the same two narrations are found in several books so they’re just being repeated.
for example the Shia they’ll quote from al-Dhurr al-Manthur by al-Suyuti and make it sound as if it’s a different narration whereas in reality this is the exact same narration of ibn Jarir al-Tabari, or they might quote it from Kanzul-‘Ummal by al-Muttaqi al-Hindi and say that this is a different source whereas in reality al-Hindi quoted it from al-Suyuti who in turn quoted it from Ibn Jarir with the exact same weak Sanad.
Other things which the Shia might use:
-They might use the narration of Ibn Katheer in al-Seerah al-Nabawiyah who took it from al-Beihaqi with its Isnad from ‘Ammar from Abi al-Zubair from Jabir and this narration doesn’t even mention ‘Umar (ra) so it’s useless.
-They might use the saying of al-Fakhr al-Razi who is an Asha’ari in his Tafseer and his saying is not backed by any proof from the Quran or the Ahadith so it’s useless.
-They might quote the saying of an Orientalist called Marsden Jones who commented on the book al-Maghazi by al-Waqidi and the sayings of all Orientalists aren’t worth anything in Islam.
-They might use a saying from the book Mu’ujam al-Buldan by al-Hamawi and this doesn’t even have a Sanad so it’s useless.
-They may quote the sayings of Ibn abi al-Hadeed who is a Shi’ite/Mu’atazili, he is weak and his sayings are ignored altogether.
Any addition by the brothers is welcome, this is the stuff I came across until now.