#I have modified the main loop query via query_posts(). Why isn't pagination working, and how do I fix it?

Great answer Chip created needs to be modified today.
For some time we have $wp_the_query variable that should be equal to the $wp_query global just after the main query executes.

This is why this the part from the Chip's answer:

###Hack the main query object

is not needed anymore. We can forget this part with creating the temporary variable.

// Pagination fix
$temp_query = $wp_query;
$wp_query   = NULL;
$wp_query   = $custom_query;

So now we can call:

$wp_query   = $wp_the_query;

or even better we can call:

wp_reset_query();

Everything other Chip outlined stays. After that query-reset-part you can call the pagination functions that are f($wp_query), — they depend on $wp_query global.

In order to further improve the pagination mechanics and to give more freedom to the query_posts function I created this possible improvement:

https://core.trac.wordpress.org/ticket/39483

I have modified the main loop query via query_posts(). Why isn't pagination working, and how do I fix it?

Great answer Chip created needs to be modified today.
For some time we have $wp_the_query variable that should be equal to the $wp_query global just after the main query executes.

This is why this the part from the Chip's answer:

Hack the main query object

is not needed anymore. We can forget this part with creating the temporary variable.

// Pagination fix
$temp_query = $wp_query;
$wp_query   = NULL;
$wp_query   = $custom_query;

So now we can call:

$wp_query   = $wp_the_query;

or even better we can call:

wp_reset_query();

Everything other Chip outlined stays. After that query-reset-part you can call the pagination functions that are f($wp_query), — they depend on $wp_query global.

In order to further improve the pagination mechanics and to give more freedom to the query_posts function I created this possible improvement:

https://core.trac.wordpress.org/ticket/39483