Gfrktyl

King David is well-known to be positively regarded in Judaism, not only for his great accomplishments and devotion (psalms, if not authored by him, became prominent because of him), but also as the predecessor of the (yet-to-come) Messiah.

Similarly, King Solomon also made great accomplishments (e.g. was the builder of the First Temple of Jerusalem), and some books of the Tanach were allegedly written by him.

However, Saul seems to be relegated to a second tier of importance. Is this really the case? For instance, liturgically speaking, is Saul positively mentioned in prayers and songs, or positively commemorated in religious holidays or celebrations?

There are not very many references to Saul in the liturgy. Here is one that portrays him somewhat negatively, in the context of his failure to destroy Amalek (from Yotser for Zachor):

זָכוֹר נָגִיד לְחָמְלוֹ הִסְכִּים. סָר מִמְּלוּכָה מִבֵּין נְסִיכִים. עוֹלֵל הַנּוֹתָר עָמַד לְשִׂכִּים: ‏

Remember the noble who agreed to have mercy on him [Agag]. He left the kingship from among princes. The remaining child [Haman the Agagite] stood up as thorns.

Other sources portray him both positively and negatively. Here are three sources that explicitly compare him with David. Two of them portray him as inferior:

Eruvin 53a-b (translation modified from link):

דוד גלי מסכתא שאול לא גלי מסכתא דוד דגלי מסכתא כתיב ביה יראיך יראוני וישמחו שאול דלא גלי מסכתא כתיב ביה (אל כל) אשר יפנה ירשיע ‏

King David would disclose the tractate to be studied in advance, whereas Saul would not disclose the tractate to be studied. With regard to David, it is written: “Those who fear You will see me and be glad” (Psalms 119:74). With regard to Saul, who would not disclose the tractate to be studied, it is written: “And wherever he turned himself he did them mischief” (i Samuel 14:47).

Bereshit Rabba 25:3 (my translation):

רַבִּי הוּנָא וְרַבִּי יִרְמְיָה בְּשֵׁם רַבִּי שְׁמוּאֵל בַּר רַב יִצְחָק עִקַּר אַוְתֶּנְטְיָא שֶׁלּוֹ לֹא הָיָה רָאוּי לִהְיוֹת בִּימֵי דָּוִד אֶלָּא בִּימֵי שָׁאוּל, אֶלָּא עַל יְדֵי שֶׁהָיָה שָׁאוּל גְּרוֹפִית שֶׁל שִׁקְמָה, גִּלְגְּלוֹ הַקָּדוֹשׁ בָּרוּךְ הוּא וֶהֱבִיאוֹ בִּימֵי דָּוִד ‏

Rabbi Huna and Rabbi Yirmiya in the name of Rabbi Shemu'el bar Rav Yitschak: The worst part of [the famine in David's time] should have come not in David's time, but in Saul's time. But because Saul was a [fruitless] sycamore branch, the Holy One Blessed is He brought it in the time of David [who was an olive branch and could survive it (words added in Rut Rabba 1:4)].

And one source that portrays him as superior to David, Midrash Tehillim 7:2 (my translation):

(שמואל-ב ג ז) ולשאול פלגש. כד איתעבד שאול מלך הוא עוד זעירא אפיך¹ וכתיב (שם ה יג) ויקח דוד עוד נשים ופלגשים מירושלים. ואת מדמה עצמך לשאול. שאול (שם א כג) מנשרים קלו (רגליו) מאריות גברו. ר' לוי אמר ששים מיל היה מהלך שאול ביום אחד. ור' סימון אמר מאה ועשרים. ורבנן אמרי מאה ושמונים. ואימתי כשנשבה הארון. הדא הוא דכתיב (שמואל-א ד יב) וירץ איש בנימין מהמערכה זה שאול ואת מדמה עצמך לשאול. אדם הולך לבית המשתה אינו מוליך בניו עמו מפני מראית העין. ושאול הולך למלחמה ויודע שמדת הדין פוגעת בו ומוליך בניו עמו ואת מדמה עצמך לשאול. ואת אומר (שמואל-ב כא יז) לא תצא עוד אתנו במלחמה. שאול אוכל חולין בטהרה שנאמר (שמואל-א ט כד) וירם הטבח את השוק והעליה. רבי יוחנן אומר השוק והאליה. רבי אומר השוק והחזה שהיה אוכל חולין בטהרה ואת מדמה עצמך לשאול. ‏

"And Saul had a concubine" (2 Samuel 3:7). When Saul became king, you were still unimportant, and yet it is written "And David took more wives and concubines from Jerusalem" (ibid 5:13), and you compare yourself to Saul?! Saul was "lighter than eagles, stronger than lions" (ibid 1:23) [...] and you compare yourself to Saul?! A person going to a feast doesn't bring his sons with him for fear of onlookers, and yet Saul goes to battle knowing that the attribute of judgment will strike him, and he brings his sons with him, and you compare yourself to Saul, while [David's soldiers] say "Don't go out to battle with us any more" (ibid 21:17)?! Saul ate ordinary food in purity, as it is said "The cook lifted up the thigh and what was on it" (1 Samuel 9:24) [...] and you compare yourself to Saul?!

¹_{Typo corrected from אפילו in accordance with this edition.}

Shaul was great beyond the imagination of the people in his Generation (and how much more so ourselves) and cannot be Judged. Yalkut Shimoni Shmuel II 157:

שגיון לדוד אשר שר לה' על דברי כוש בן ימיני (Tehilim,9,1),

א"ל הקב"ה לדוד דוד שירה אתה אומר לפני על מפלת של שאול אלמלא אתה שאול והוא דוד אבדתי כמה דוד מפניו והיינו דכתיב שגיון לדוד אשר שר לה' על דברי כוש בן ימיני. כתיב כוש והלא שאול שמו, אלא מה כושי משונה בעורו אף שאול משונה במעשיו ובמראיו שנאמר משכמו ומעלה גבוה מכל העם

When Dovid was being chased Hashem Told him if Shaul had Dovid's Mazel and Dovid had Shaul's Mazel Shaul would have killed many people like Dovid i.e Shaul was the greatest man at the time as it says "Head and Shoulders above everyone" (even more great than Dovid at the time) but because he had a bad Mazel he couldn't remain the King and Dovid had to take over the Kingship.

Shaul was forgiven for all his sins and had a Place on the same level as Shmuel Hanavi in the world to come Brachos 12b:

ומנין דאחילו ליה מן שמיא שנא' (שמואל א כח, יט) (ויאמר שמואל אל שאול) [ו] מחר אתה ובניך עמי וא"ר יוחנן עמי במחיצתי

King saul is regarded as not so positive figure as he disobeyed a direct order to eliminate all Amalek.

Instead he spared Amalek's king and by doing that he allowed Haman to be born later.
Samuel the prophet was furious and said that his kingdom will be torn apart and as a result Saul lost his sanity and even tried to kill David by throwing a spear at him.

King saul also used an oracel to bring Samuel from the dead which is strictly vorbidden by moses's laws.
But a king is a king and there is a beautiful Rashi comment on this which says that when the oracel brought people from the dead they were hanging upside down and samuel came normal becuase of king's respect according to moses's laws.

That's why the oracel was frightend in the story and knew that he was King saul.

He also commited suicide on the battle and dropped on his sword which is also vorbidden.

But, king saul was elected because the people wanted a king and he was crowned by force by Samuel in a direct order from god so I can't blame him for anything as he didn't want all this.