Notify is all you get right now.  That's what's causing Windows to put the beep and message text bubble.  APRSIS32 doesn't do anything expect fire off a Windows notification, and only if you have Notify checked as you've observed.  In most situations that I've observed, bulletins are set and forgotten such that they periodically re-transmit themselves.  It was due to that "feature" that I made it possible to turn OFF the notification because it fires way to often in some circumstances.

What version of Windows are you running?  I believe Windows 10 keeps the notifications until you clear them, but it does auto-hide them.  Check the icon to the extreme right (or bottom) end of your task bar.  Clicking that should bring up all notifications that you haven't explicitly dismissed.  I must confess, however, that I'm not positive that APRSIS32's Bulletin notifications will show up there.

It's the "Action Center" button and looks like this (at least on my machine) and lives just to the right of the date/time.

Lynn (D) - KJ4ERJ - Author of APRSISCE for Windows Mobile and Win32

On 10/9/2020 5:32 PM, Greg D wrote:

Hi Lynn,

So I've noticed recently that I'm getting an occasional Bulletin message
sent out by someone (KM6HBH) who's running a net a little ways away. 
Totally legit, nice to know; not complaining about that.

But when received, the Bulletin creates a single "Beep" on the computer,
and a message text bubble in the upper right corner of the screen which
goes away on its own a few seconds later.  If I happen to be nearby and
hear the beep, and then catch the bubble before it goes away, I can see
it.  Otherwise, there's no hint that it even happened.   That same Beep
is also generated when the Internet goes out, which, unfortunately, can
be a similarly frequent event, so it can be ignored due to fatigue at times.

I see under Messages / Bulletins / View that I can check to see what's
been received, but that kind of defeats the time-value of the Bulletin
message.  Is there any way to configure how a Bulletin is presented on
the screen, e.g. for how long, and with what sort of notification flag
(like the Messages rectangle in the upper right corner of the map)? 
There's a check for Notify in the Messages / Bulletin menu, but no
Configure.

Thanks,

Greg  KO6TH