Which countries (if any) allow beyond-line-of-sight drone use?
The current situation (November 2018) regarding beyond visual line of sight operations (BVLOS) is heterogeneous:


Overall, not allowed in Europe, but there are notable exceptions where BVLOS is explicitly regulated, see below.
Regulated in Canada
Regulated in a few countries in Asia, Africa and Latin America.
Not allowed in the US


US

UAVs fall under Part 107 operations which prevents BVLOS (107.31). Waivers can be requested, but only 27 have been granted to date (enter "107.31" in the search box).

European Union or EASA member states

EASA is the European safety agency. For EASA member states (EU and a few other countries), there are two opposite situations today:

With no surprise North European countries (Iceland, Norway, Sweden), with many isolated locations have allowed BVLOS operations early, including for delivery of goods to consumers, e.g. Aha, an online marketplace, delivers goods using drones in Iceland. Not only this is BVLOS, but it's even automated.

Many countries don't allow it but they are waiting for a EU regulation that would allow them. Actually current opinion of the EASA is it must be allowed (but more for industrial uses than for good delivery to public):


  EASA will develop standard scenarios that will make it simpler to
  obtain authorisations for well‑defined operations (such as, for
  example, linear inspections conducted in BVLOS, or crop spraying).


(Source)

In their declaration EASA mentions linear inspections, which has been actually something done in France since a long time for railways catenaries and electric grid lines.

France has kept with its pioneering role in aviation. BVLOS is regulated since 2012. A company located in Toulouse, another well known place for aviation, paved this road. As I'm not going to do their advertising, I let you read about them online, e.g. last year they were able to have a 50-km civilian flight with a 3G data link.

Switzerland

If you are asking for your home country, Switzerland, which is also an EASA member state:


  


LOS is required, but waivers are possible, though there are no public criteria provided, so you need to try, when you have defined all the details of your project.

As DeltaLima mentioned (maybe with a good insider knowledge) laws are rapidly evolving in the domain, as does UAV technology.

Around the World

Taken from International Commercial Drone Regulation and Drone Delivery Services by Rand Corporation:


  For countries that have drone regulations, these regulations typically
  incorporate a pilot’s license, registration of the drone, restricted
  zones, and insurance; regulation requirements vary based on such
  parameters as drone mass, population density, altitude, and use cases.


and the list of countries in 2017:


  


As visible there are legal possibility for BVLOS in many countries and across all continents.

Permission limits

Same document, excerpt for three countries:


  


For France already mentioned, we see the BVLOS is authorized up to 1 km for 2 kg+ mass, and unlimited below (but constraints like pilot license and registration still apply). Looking at the criteria for a permissive country:


  


Well, except height and insurance in certain cases, there is no other ones!

Future market

While B2B application have a large market, the B2C market is huge in an era of online sales and home delivery. Amazon is waiting for (experimentation in Japan), as quick food businesses do, for their delivery.