For home service industry - what are the best discounts to offer that will attract the most customers?

I would update the question a bit to specify the target market, specific audience, and market fit. What specific home service are you looking to target? Maid cleaning service is very different from plumbing, electrician, or roof and gutter services. I recommend asking a better question that will help narrow down you market and service industry. The answer in this case really depends on the specific industry, customer segment, target audience, target market, demographic, and household annual income.

I also recommend focusing on your Unique Selling Proposition and Value Proposition that you offer to your customers. Discounts are great but customers will choose home service companies based on the following factors: honesty, integrity, reliability, quality of service, customer service, attention to detail, and follow through after the service is completed.

You can't beat these factors with "best discounts". Many customers have experienced poor service in the past and will choose premium, reliable, quality service and customer service over price each time. A home service business that is reliable and does the work they were hired to do with attention to detail and transparent communication will provide customers that peace of mind and reassurance they hired the right company.

Agreed here. What is your target? One off sell, sure a discount will help the consumer to pull the trigger. But what about engagement and long-term relations? 

Discounts are great but no obligation engagements are best! Offer a small discount and communicate clearly that you can try the service once and cancel with no hassle if you choose not to proceed.

Free estimate with a discount percentage if customer decides to proceed. Price matching to competitors.