At the top of the SharePoint Home page (_layouts/15/sharepoint.aspx) is a search box. The items shown as a result of using that search box do not respect any configured Display Templates.

When sharing content in SharePoint Online, the sharing interface allows a user to type the name of an Office 365 Group. SharePoint Online content that is shared in this manner will not appear in SharePoint Search results to the members of the Office 365 Group. This doesn't provide a consistent experience for users, as many users may not be aware what entries are Office 365 Groups, Azure Active Directory security groups, etc. We'd like to see content shared to the members of an Office 365 group in this manner appear in the search results for those users.

So we can show search refiners for filtering results based on library name.

Currently, a user can only search the site if the user knows the name of the site or part of the name.
When a new site is created a lot of times users doesn't remember the name. Also, the site is not listed in the frequent or suggested sites.
I think, it would be useful if a list is displayed which contains all the sites that user has access to across the farm (not limiting to a particular site collection).

For manual download the OOTB SharePoint reports from SP admin page- SharePoint Online admin portal> Search> Search Reports.
Multiple reports available OOTB, but no API/Powershell available for Search Analytics, by which we can export custom Search reports or we can schedule the download of OOTB reports Monthly.

Provide API/Powershell for managing/automating SharePoint Search reports.

Enable some mechanism that allows a consumer to provide feedback on the results they get in search (e.g. not getting expected results or feedback tied directly to a specific content item). General feedback would be routed to one place while content item level feedback would be automatically routed to the assigned author.

This idea is to create a SharePoint Online search connector to Azure Data Lake. This should be similar to the file share connector we have in hybrid search, but connect directly from the SharePoint Online search engine to Azure Data Lake. This will allow us to include files stored in Azure Data Lake in the enterprise search solution.

Wanted to display items when certain field is not empty.
If possible please share us why it's not implemented?

Integrate Delve Search results with SharePoint search center

Currently we have a large amount of file share and SharePoint data that contains Personally Identifiable Information and Sensitive Data. It would be ideal to be able to auto tag documents at the index level to filter out those documents in Search results. Additionally, it would also allow for tagging documents in a positive way, for more refined searching.

Make it easier to add/remove servers from the Search Service Application.

Using the search at the top left of the site contents page has a filtering impact on the contents and subsites listings. This functionality is a complete divergence from the search experience in other parts of SharePoint. There should be a way to send all search boxes to the same search results page, instead of this rather confusing setup.

Allow to add custom file extension type like .DWG etc. so that search results could be able to link to actual file like .pdf etc.

I worked with Microsoft support to create a new search scope that included the wildcard for every entry at the top level site. We applied this template as the default and it works when searching at the top level site. However, it does not apply when searching within a Document library that resides at the same level as the parent. Ultimately, we do not want to require a wildcard for every partial search. If someone wants to find instructions in our library for GoToMeeting they should be able to find it by simply entering Goto.

Currently in the SharePoint Online (SPO) tenant, customising feature for word-breaking dictionary is not offered.
This is one of the painpoints for tenant administrators because amount of the data in the tenant is growing up day by day.

Actually the customization feature for word-braking is offered in SharePoint Server (On premise) but it shold also be offered to the users who are using SPO.

The spreadsheet generated from the usage report link under Popularity Trends generates incorrect statistics for unique users in the section that is aggregated for monthly statistics. The spreadsheet appears to sum up the unique users for each day in the month, rather than calculating the unique users across the entire month. This will tend to greatly exaggerate the unique users per month.

The setting "Always index all Web Parts on this site", is not yet exposed trough CSOM.

This setting can be found in site settings > Search and Offline Availability > Indexing ASPX Page Content

This is usefull when using sites with broken permissions, which are then not indexed as needed.

Please add this functionality to CSOM.
It is this property: web.ASPXPageIndexMode = WebASPXPageIndexMode.Always;