Below you find all the available assignments with the respective settings for each.

Menu Items

In addition to the core Menu Items selection, it's possible to automatically include or exclude all current and future menu items placed under a parent menu item.

Selection Select the menu items to assign to.
Also on child items Also assign to child items of the selected items?
Include no Itemid Also assign when no menu Itemid is set in URL?

Home Page

Unlike selecting the home page (default) item via the Menu Items, this will only match the real home page, not any URL that has the same Itemid as the home menu item.

This might not work for all 3rd party SEF extensions.

Date & Time

The date and time assignments use the date/time of your servers, not the ones of the visitor's system. 

The date range can be set to be recurring every year in the same time period.

Date

Start Publishing Enter the date to start publishing
Finish Publishing Enter the date to end publishing
Recurring Select to apply date range every year. (So the year in the selection will be ignored)

Seasons PRO

Selection Select seasons to assign to.

Options: Winter, Spring, Summer, Fall / Autumn
Hemisphere Select the hemisphere your website is located in

Options: Northern, Southern, Australia

Months PRO

Selection Select months to assign to.

Options: January, February, March, April, May, June, July, August, September, October, November, December

Days of the week PRO

Selection Select days of the week to assign to.

Options: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday

Time PRO

Start Publishing Enter the time to start publishing.
Format: 23:59
Finish Publishing Enter the time to end publishing.
Format: 23:59

Languages

The Languages assignment differs from the core in that it allows a selection of multiple languages.

Selection Select the languages to assign to.

Joomla! Content

This assignment is based on what Joomla content page is displayed, including Page Types, Categories, and specific Articles.

Page types

Selection Select on what page types the assignment should be active.
Options: Archived Articles, Single Article, List All Categories, Category Blog, Category List, Featured Articles, Create Article

Categories

Selection Select the categories to assign to.
Also on child items Also assign to child items of the selected items?
Page types Select on what page types the assignment should be active.
Options: Categories, Articles, Others

Articles

Selection Select the articles to assign to.
Featured Select to use the feature state in the assignment.
Date Select the type of date comparison to assign by:
Options: Created Date, Modified Date, Start Publishing, Finish Publishing
Comparison Methods: From-To, Before, After
Authors Select the authors to assign to.
Content Keywords Enter the keywords found in the content to assign to. Use commas to separate the keywords.
Meta Keywords Enter the keywords found in the meta keywords to assign to. Use commas to separate the keywords.
Custom Fields Select the custom field(s) you want to assign to and enter the desired value(s).
Comparison Methods: Equals, Is not equal to, Contains, Contains one of, Does not contain, Begins with, Ends with, Less than, Greater than, Regular Expressions

For complete details on how to use the Custom Fields assignments, the Comparison Methods, and even how to use Relative Dates as part of your checks, see the dedicated Blog Post.

Tags

Selection Enter the tags to assign to. Use commas to separate the tags.
Match all Select to only let the assignment pass if all of the selected items are matched.
Also on child items Also assign to child items of the selected items?

User Group Levels

Selection Enter the user groups to assign to.
Match all Select to only let the assignment pass if all of the selected items are matched.
Also on child items Also assign to child items of the selected items?

Users PRO

Selection Select the users to assign to.

Components

Selection Select the components to assign to.

URL

The URL assignment allows control over specific pages or groups of pages, with support for Regular Expressions.

URL matches Enter (part of) the URLs to match.
Use a new line for each different match.
Use Regular Expressions Select to treat the value as regular expressions.

URL parts will be matched using regular expressions. So make sure the string uses valid regex syntax.
Case Sensitive Url parts will be only match if casing is exactly the same.

Devices

Selection Select the devices to assign to. Keep in mind that device detection is not always 100% accurate. Users can setup their device to mimic other devices.

Options: Desktop, Tablet, Mobile

Operating Systems

Selection Select the operating systems to assign to. Keep in mind that operating system detection is not always 100% accurate. Users can setup their browser to mimic other operating systems.

Browsers

Browsers Select the browsers to assign to. Keep in mind that browser detection is not always 100% accurate. Users can setup their browser to mimic other browsers
Mobile Browsers Select the browsers to assign to. Keep in mind that browser detection is not always 100% accurate. Users can setup their browser to mimic other browsers

IP Addresses PRO

IP Addresses / Ranges A comma and/or enter a separated list of IP addresses and IP ranges. For instance:
127.0.0.1
128.0-128.1
129

Geolocating PRO

Geolocating is not always 100% accurate. The geolocation is based on the IP address of the visitor. Not all IP addresses are fixed or known.
To be able to use this assignment, you need to install the Regular Labs GeoIP library.

Continents Select the continents to assign to.
Countries Select the countries to assign to.
Regions / States Select the regions / states to assign to.
Postal Codes A comma separated list of postal codes (12345) or postal code ranges (12300-12500).

Custom PHP PRO

This gives you the opportunity to assign to just about anything you can think of. You just need to know what PHP code you should use.

With PHP you can call on all data/information available in the variables, URL, database, etc. And then do your checks on this data. You can tell the control the display by ending the PHP code with a return true or false based on your checks.

For examples and ready-to-use PHP scripts, check out the "Using Custom PHP Conditions" guide.

Custom PHP Enter a piece of PHP code to evaluate. The code must return the value true or false.

For instance:
return ( $user->name == 'Peter van Westen' );

3rd party Content PRO

EasyBlog

Page Types

Selection Select on what page types the assignment should be active.

Options: Archive Layout, Bloggers Layout (All authors), Categories Layout (All categories), Overview Layout (Dashboard), Single Blog Entry Layout, Featured Blogs Layout, Frontpage Blog Layout, Login layout, My Blog Layout, Subscriptions Layout, Tags Layout, Team Blog Layout

Categories

Selection Select the categories to assign to.
Also on child items Also assign to child items of the selected items?
Page types Select on what page types the assignment should be active.

Options: Categories, Items

Tags

Selection Enter the tags to assign to. Use commas to separate the tags.
Page types Select on what page types the assignment should be active.

Options: Tags, Items

Items

Selection Select the items to assign to.
Authors Select the authors to assign to.

FLEXIContent

For category & article (item) assignments, see the above Joomla! Content section.

Page types

Selection Select on what page types the assignment should be active.

Options: Author, Category, My Items, Favourites, Directory, Single item., Search, Tag

Tags

Selection Enter the tags to assign to. Use commas to separate the tags.
Page types Select on what page types the assignment should be active.

Options: Tags, Items

Types

Selection Select the types to assign to.

Form2Content Lite

For category & article (item) assignments, see the above Joomla! Content section.

Types Select the types to assign to.

K2

Page types

Selection Select on what page types the assignment should be active.

Options: Item, Item edit form, Categories, Tag, User page (blog), Latest items from (one or more) users or categories

Categories

Selection Select the categories to assign to.
Also on child items Also assign to child items of the selected items?
Page types Select on what page types the assignment should be active.

Options: Categories, Items

Tags

Selection Enter the tags to assign to. Use commas to separate the tags.
Page types Select on what page types the assignment should be active.

Options: Tags, Items

Items

Selection Select the items to assign to.
Content Keywords Enter the keywords found in the content to assign to. Use commas to separate the keywords.
Meta Keywords Enter the keywords found in the meta keywords to assign to. Use commas to separate the keywords.
Authors Select the authors to assign to.

ZOO

Page types

Selection Select on what page types the assignment should be active.

Options: Category, ITEM, Frontpage, My Submissions, Submission

Categories

Selection Select the categories to assign to.
Also on child items Also assign to child items of the selected items?
Page types Select on what page types the assignment should be active.

Options: Categories, Items

Items

Selection Select the items to assign to.
Authors Select the authors to assign to.

HikaShop

Page types

Selection Select on what page types the assignment should be active.

Options: User addresses, Categories listing, Featured Articles, Create Article, Contact page, Products listing, Product page, User control panel, Registration form

Categories

Selection Select the categories to assign to.
Also on child items Also assign to child items of the selected items?
Page types Select on what page types the assignment should be active.

Options: Categories, Products

Products

Selection Select the items to assign to.

MijoShop

Page types

Selection Select on what page types the assignment should be active.

Options: Account, Admin Panel, Affiliates, Bestsellers, Shopping Cart, Checkout, Compare, Contact, Downloads, Home, Latest Products, Login, Manufacturers, Newsletter, Orders, Popular Products, Registration, Returns, Search, Sitemap, Specials, Voucher, Wishlist

Categories

Selection Select the categories to assign to.
Also on child items Also assign to child items of the selected items?
Page types Select on what page types the assignment should be active.

Options: Categories, Products 

Products

Selection Select the items to assign to.

RedShop

Page types

Selection Select on what page types the assignment should be active.

Options: Account, My Wishlist, Cart, Catalogue, Sample, All Products From a Category, Categories, Category Detailed, Checkout, Gift Cards List, Login, Logout, Portal Layout For Shopper Group, Portal Detail Layout For Shopper Group, Manufacturers, Manufacturer Details, Products From Selected Manufacturer, Newsletter Subscription, Orders, Orders Tracker, Reset Password, Set Password, Token, Product, Product Download, Product Rating, Quotation, Registration, Product Search, Newest Products, Products On Sale, redFILTER, My Wishlist, All Wish Lists

Categories

Selection Select the categories to assign to.
Also on child items Also assign to child items of the selected items?
Page types Select on what page types the assignment should be active.

Options: Categories, Products

Products

Selection Select the items to assign to.

VirtueMart

Page types

Selection Select on what page types the assignment should be active.

Options: Shopping Cart, VirtueMart Categories Layout, Category Layout, Manufacturer Default Layout, Manufacturer Details Layout, List Orders, Product Details Layout, Account Maintenance, User Edit Address, Displays vendor contact, Displays vendors, Displays vendor details, View vendor TOS, Front page

Categories

Selection Select the categories to assign to.
Also on child items Also assign to child items of the selected items?
Page types Select on what page types the assignment should be active.

Options: Categories, Products

Products

Selection Select the items to assign to.