Sounds like the Desktop is configured as POP and the laptop and tablet use IMAP. If you want to use multiple devices they should all use IMAP or tell the POP devices to leave the messages on the server.

No, all are IMAP and are marked not to delete messages. Still investigating. Cheers

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Info is as follows:

 Application Basics

   Name: Thunderbird
   Version: 38.4.0
   User Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101 Thunderbird/38.4.0
   Profile Directory: Open Directory

             (Local drive)
   Application Build ID: 20151125201005
   Enabled Plugins: about:plugins
   Build Configuration: about:buildconfig
   Memory Use: about:memory

 Mail and News Accounts
   account1:
     INCOMING: account1, , (imap) imap.netspace.net.au:143, plain, passwordCleartext
     OUTGOING: smtp.netspace.net.au:25, alwaysSTARTTLS, passwordCleartext, true

   account2:
     INCOMING: account2, , (none) Local Folders, plain, passwordCleartext

 Extensions
   Messaging Menu and Unity Launcher integration, 1.3.1, true, messagingmenu@mozilla.com

 Important Modified Preferences

   Name: Value

     browser.cache.disk.capacity: 358400
     browser.cache.disk.smart_size_cached_value: 358400
     browser.cache.disk.smart_size.first_run: false
     browser.cache.disk.smart_size.use_old_max: false
     extensions.lastAppVersion: 38.4.0
     mail.openMessageBehavior.version: 1
     mailnews.database.global.datastore.id: 546df262-db32-46ff-b890-93e875fb365
     network.cookie.prefsMigrated: true
     network.predictor.cleaned-up: true
     places.database.lastMaintenance: 1450260181
     places.history.expiration.transient_current_max_pages: 101915
     plugin.importedState: true
     print.print_bgcolor: false
     print.print_bgimages: false
     print.print_colorspace: default
     print.print_downloadfonts: false
     print.print_evenpages: true
     print.print_in_color: true
     print.print_margin_bottom: 0.5
     print.print_margin_left: 0.5
     print.print_margin_right: 0.5
     print.print_margin_top: 0.5
     print.print_oddpages: true
     print.print_orientation: 0
     print.print_page_delay: 50
     print.print_paper_data: 0
     print.print_paper_height: 279.40
     print.print_paper_name: na_letter
     print.print_paper_size_type: 1
     print.print_paper_size_unit: 1
     print.print_paper_width: 215.90
     print.print_plex_name: default
     print.print_resolution_name: default
     print.print_scaling: 1.00
     print.print_shrink_to_fit: true
     print.print_to_file: false
     print.print_unwriteable_margin_bottom: 56
     print.print_unwriteable_margin_left: 25
     print.print_unwriteable_margin_right: 25
     print.print_unwriteable_margin_top: 25

 Graphics

     Adapter Description: NVIDIA Corporation -- GeForce GT 330M/PCIe/SSE2
     Vendor ID: NVIDIA Corporation
     Device ID: GeForce GT 330M/PCIe/SSE2
     Driver Version: 3.3.0 NVIDIA 340.96
     WebGL Renderer: false
     GPU Accelerated Windows: 0

     AzureCanvasBackend: cairo
     AzureSkiaAccelerated: 0
     AzureFallbackCanvasBackend: none
     AzureContentBackend: cairo

 JavaScript

 Incremental GC: 1

 Accessibility

   Activated: 0
   Prevent Accessibility: 0

 Library Versions

     Expected minimum version
     Version in use

     NSPR
     4.10.10
     4.10.10

     NSS
     3.19.2.1 Basic ECC
     3.19.2.1 Basic ECC

     NSS Util
     3.19.2.1
     3.19.2.1

     NSS SSL
     3.19.2.1 Basic ECC
     3.19.2.1 Basic ECC

     NSS S/MIME
     3.19.2.1 Basic ECC
     3.19.2.1 Basic ECC