The Scottish government's 2013 white paper on independence - Scotland's Future - sets out a number of tangible assets that, the paper argues, Scotland would be entitled to a proportion of based on population share. For example, an independent Scotland would seek to take ownership of a share of the United Kingdom's overseas properties, e.g. embassies & consulates:
Scotland will be entitled to a fair share of the UK's extensive
overseas properties (or a share of their value) allowing us to use
existing premises for some overseas posts. For example, the Foreign
Office owns or leases almost 5,000 properties overseas [source]. The
estimated value of this estate is around £1.9 billion. Based on a
population share (our actual share will be a matter for negotiation)
Scotland would be entitled to around £150 million allowing us to
establish ourselves quickly and for little initial cost in our
Chapter 6 - International Relations and Defence
The paper also argues that Scotland would be entitled to £7.8 billion of the UK's existing defence assets - based on a population-share proportion of the 2007 valuation of ~£93 billion performed by the Ministry of Defence. It even splits out what particular assets Scotland would seek from the UK, including, amongst others, "one aviation unit operating six helicopters", "a Quick Reaction Alert (QRA) squadron incorporating a minimum of 12 Typhoon jets", "two frigates from the Royal Navy's current fleet" and so on.
Other assets mentioned in the report that an independent Scotland would seek to inherit include BBC Scotland, "assets that are not related directly to particular services, such
as the UK’s public shareholdings in banks", "bespoke IT software", and so on.
Page 21 of the report does mention that Scotland may choose to offset its share of national debt against these assets.
There are also implications for the United Kingdom's nuclear deterrent - Trident. The white paper makes clear that an independent Scotland would no longer accept Trident being based at the ports of Faslane & Coulport:
The Scottish Government is committed to securing the complete
withdrawal of Trident from an independent Scotland as quickly as can
be both safely and responsibly achieved.
As the rest of the UK has no bases outside of Scotland suitable for Trident, this would present a significant problem for the Westminster government. Indeed, it has been proposed more recently that Scotland should fund its defence by leasing the bases out while a replacement is constructed - something which would almost certainly prove to be extremely expensive for the UK taxpayer. In a worst case scenario, the loss of the UK's nuclear deterrent would lead to a further reduction in its international standing. For example, the UK would become the only permanent member of the UN Security Council without nuclear capabilities.