Why didn't Hela use Odin's spear to open the Bifrost?

1. She did not have it.

   This is the most likely reason because we don't actually see much of Odin's spear but the times we do, we see Thor is seen wielding it.

   According to Cinematic Wiki

   When Hela came to take the throne of Asgard, Thor had taken the spear, and briefly used it in a duel before dropping it. As he seems to have left it after awakening his true power, and the whole of Asgard was subsequently destroyed by Surtur in Ragnarök, Gungnir seems to have been destroyed.

   Plus when Thor actually drops Odin's spear she isn't focused on opening the bifrost because she realizes she has to defeat Thor before she can leave Asgard. If she wanted to leave Asgard so much she would have focused her troops/Fenrir to attack only Heimdall.

2. Perhaps she does not know it can open the Bifrost.

   Ok as you say "she knows of its existence but she has been locked up by Odin for centuries. Maybe the fact that Odin's spear can open up the bifrost is something new to her and not done before.

   Imagine you're Hela and you just returned to Asgard after centuries of imprisonment, when she returns to Asgard she sees that it's the sword that opens the bifrost and to her knowledge she probably assumes that's the only thing that opens the bifrost. We cannot assume she knows that Odin's spear opens the bifrost. Obviously she knows its a powerful weapon but she doesn't need weapons at that point she needs a key.

3. Thor and Loki are stuck outside Asgard/presumed dead, the warriors three are actually dead and so is most of the Asgardian hierarchy.

   Its unlikely that the people of Asgard themselves know where the location of the spear is never mind how to use it.

   We can't assume that Hela doesn't know how to wield Odin's spear but the people that know how to (Thor and Loki) are not in Asgard.

4. It's better for Hela to find and obtain the sword than to look for Odin's spear.

This is because by finding the sword she would be killing two birds with one stone. Hela finds out that someone has stolen the sword trapping herself and her army in Asgard and this means someone is opposing Hela.

It would be better for her to find and destroy the rebels whilst regaining the sword. She might be more angry that someone stole from her rather than the fact she can't leave Asgard.

So, why not destroy the rebel Asgardians whilst also getting Heimdall's sword?

Comments on your edits:

In the decor that Hela shows on the dome in hall(where Hela is shown to wield Mjolnir), Odin is show to wield spear, so she is aware about its existence.

She is aware of its existence but she might not be aware of its powers. More specifically its ability to open the bifrost.

At the start of Ragnarok, Loki had the spear and when Loki and Thor visits Earth to get Odin, I assumes spear is in Asgard.

It's fair to assume it's in Asgard but Hela however, doesn't know that. Also as I mentioned previously the only people that would probably know were either presumed dead or unknown.

Hela was Odin's executioner and had participated in lots of war. If Loki can find out what Spear can do, so can Hela. In fact, there is a higher possibility as when in war, you tend to use more abilities of a weapon. Considering this as opinionated comment, would not stress on it.

This is kind of a big assumption but then again you must remember Asgard was just recently built. Hela wanted to keep on conquering other realms but Odin refused and therefore banished Hela. Asgard was created after Odin and Hela conquered the 9 realms.

As a king, considering the might of spear, you would not keep it lying around somewhere. If Loki would have wielded it, we could assume that he put some spell to make it invisible, but since Thor was able to find out, we can assume it was somewhere in the palace. Now the return of Thor was not a short duration. Not sure but Heimdal says in a dialog: Hela's army of dead soldiers are searching for weeks. So Hela must have had enough time to look around.

True but Odin didn't really have a say on "where to keep it lying around", remember Loki trapped Odin on Earth and took the throne by impersonating him. Hela is also probably looking for these rebels and probably missed what was right under her nose.

Also, whats to say that Odin's spear can actually open the new bifrost bridge. If you remember at the end of Thor, the bifrost bridge was completely destroyed. Maybe it was rebuilt and only Heimdall's sword could open it. Having one key is a lot safer than having two keys.

Metro Boomin offered some nice answers, and I am no expert, but I THINK if she uses the spear to activate the Bifrost, she can not take it with her, and while she is quite powerful without it, she'd probably like to take it with her.

I do have a question to Metro Boomin though:

"This is kind of a big assumption but then again you must remember Asgard was just recently built. Hela wanted to keep on conquering other realms but Odin refused and therefore banished Hela. Asgard was created after Odin and Hela conquered the 9 realms."

Why do you feel Asgard was just built? Was that mentioned? Or is that from the comics? I ask, because in Thor: Dark World, we see a flashback of Bor fighting the Dark Elves (without Odin), and they were regarded as Asgardians at that point. (With the same technology they have thousands of years later...seriously, these people are the very definition of a stagnant society that should never have flourished.