In the home-screen there is 1 search bar available. If you want more search bars click at the right on Show Classic Interface.
You can get your whole search history via Show Search History.
Via Show All Fields you wil receive an explanation of the field codes (like any, au, inst, etc.) with which you can search via the Standard and Advanced tabs. If you click on a field code, it will appear in the search bar and you can put the search term between the round brackets. The Syntax tab explains Wrappers such as quotes or parentheses and Operators such as AND, OR and NOT.
The Publications tab gives you a wide range of search options. You can filter your search by Publication Type: All, Books and Journals. Below is a brief explanation of the different fields you can use:
Boolean operators : AND, OR, NOT: searches in MathSciNet insert Boolean AND between terms by default.
Click on the arrow near Author and all fields to search with will show up.
Author: surname,(comma) initial * or MR Author ID, e.g. Landsman, N* (where * stands for 1 or more characters) or 232479 (the MR Author ID).
Title: A search for the following words in the title – Constraints on determinism* – will give you all the results with those exact words (‘Constraints on determinism’) in the title, and not, for example, ‘new constraints on determinism’.
Review Text: The entire review text can be searched by entering keywords or phrases, as well as TeX codes for mathematical symbols, e.g. searching for ‘h 3’ will give you ‘(H)=3’.
Journal: search by journal title, e.g. Bulletin of the American Mathematical Society
Series: Titles of series, e.g. Contemporary Mathematics
MSC Primary: only primary classification codes, e.g. 62N10
MSC Secondary: all secondary classification codes in the Mathematics Subject Classification, e.g. 62 (Statistics), 62N (Engineering Statistics) and 62N10 (Quality Control). You cannot use wildcards *. MSC Primary 03C90; Secondary 03-02.
MR Number: a unique 7-digit number, e.g. MR2584951 (MR Author ID tab). Every author’s name in MathSciNet is allocated an MR Number, including the different forms of the name.
DOI: search for a Digital Object Identifier
Reviewer: reviewer’s name, e.g. Orlovskii, E.
Anywhere: search the entire MathSciNet, i.e. all 11 available searchable fields
When you click the Authors tab a search bar appears. Fill in the authors name: Landsman and a list of author names appears. Select one and click on the loupe.
Or click on Show All Fields and the next fieldcodes appear: au:, auid: and vn:, Click on the fieldcode au: and in de search bar appears: au: ""
Click between the quotation marks and fill in an author name and a list of author names appears. Select one and click on the loupe.
The next result appears with information about his MR Author ID, Earliest indexed Publication, Total publications, Collaboration Distance, etc.
And also his names he published under and the Classifications, Coauthors and a list of Publications, Publications by Journal and Publications by Series.
When you click the Journals tab a search bar appears. Fill in a part of a journal name: mathematical and a list of journal names appears. Select one and click on the loupe.
Or click on Show All Fields and the next fieldcodes appear: issn:, j: p: and st:. Click on the fieldcode j: and in de search bar appears: j:()
Click between the brackets and fill in a part of a journal name and a list of journal names appears. Select one and click on the loupe.
Select the journal name: Duke Mathematical Journal and a page with all the information about the journal appears, like Journal Details, Recent Issues, Journal TItle History, Mathematical Citation Quotient, Citations, Publications per Year, etc...
When you click the Series tab a search bar appears. Fill in a part of a series name: mathematical and a list of series names appears. Select one and click on the loupe.
Or click on Show All Fields and the next fieldcodes appear: issn:, p:, sn: and st:. Click on the fieldcode se: and in de search bar appears: se:()
Click between the brackets and fill in a part of a series name and a list of series names appears. Select one and click on the loupe.
Select the series name: Oxford Mathematical Monographs and a page with all the information about the series appears, like Series Details, Recent Volumes, Sereies TItle History, Mathematical Citation Quotient, Citations, Publications per Year, etc...
If you click Search MSC the next screen appears with the Mathematics Subject Classification 2020.
In the search bar you can fill in a classification like a keyword or a phrase or a 2-, 3-, or 5-digit classification.
Here you see an example of a search for a 2-digit classification 05 and their description.
When you click Show Historical Classes some classifications expand in more (red) classifications.
If you click Download PDF you get the whole Mathematical Subject Classification Scheme from 2020.
If you've done a search the Search Results appear. Click the Filters button and you get all the options to refine your results or click de Newest button to get the different sort options.
When you click Export you can select the references you want to export in BibTeX, AMSRef or EndNote format and click Get Citations to receive them.
Click on the MR-number or the title to get more information about the result.
Reviewed means that the article has been reviewed as you can see in the Review colum. There are more Review Status, like: Summary, Indexed, DML, Prelim, Expansion and Pending.
All blue text is clickable. To get the fulltext of an article click 
and than choose 
or doi.org to see if the fulltext is available.
Click cite to receive a format like BibTeX, Permalink, AMSRef or EndNote.and copy it.
You als see the Classifications and the citations if they are available.
